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G = C42.367C23order 128 = 27

228th non-split extension by C42 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.367C23, C4⋊C4.239D4, C4⋊C4.90C23, C22⋊C4.79D4, C2.21(Q8○D8), (C4×C8).350C22, (C2×C4).335C24, (C2×C8).456C23, C4.SD1640C2, C23.453(C2×D4), C4⋊Q8.110C22, (C2×Q8).91C23, C82M4(2)⋊38C2, C2.34(D4○SD16), C4.46(C4.4D4), (C2×D4).103C23, C8⋊C4.168C22, C23.38D436C2, (C22×C8).462C22, C4.4D4.32C22, C22.595(C22×D4), C42.C2.16C22, D4⋊C4.132C22, C23.41C237C2, (C22×C4).1033C23, Q8⋊C4.203C22, C23.24D4.12C2, C23.36D4.14C2, C22.16(C4.4D4), (C22×Q8).301C22, C42.78C2226C2, C42.28C2230C2, C42.30C2217C2, C42⋊C2.140C22, (C2×M4(2)).372C22, C23.38C23.13C2, C4.44(C2×C4○D4), (C2×C4).137(C2×D4), (C2×Q8⋊C4)⋊58C2, C2.46(C2×C4.4D4), (C2×C4).490(C4○D4), (C2×C4⋊C4).625C22, (C2×C4○D4).150C22, SmallGroup(128,1869)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.367C23
C1C2C4C2×C4C22×C4C22×C8C82M4(2) — C42.367C23
C1C2C2×C4 — C42.367C23
C1C22C42⋊C2 — C42.367C23
C1C2C2C2×C4 — C42.367C23

Generators and relations for C42.367C23
 G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b-1, e2=a2b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, cbc-1=b-1, bd=db, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >

Subgroups: 340 in 186 conjugacy classes, 92 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4×C8, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C22×C8, C2×M4(2), C22×Q8, C2×C4○D4, C82M4(2), C2×Q8⋊C4, C23.24D4, C23.36D4, C23.38D4, C4.SD16, C42.78C22, C42.28C22, C42.30C22, C23.38C23, C23.41C23, C42.367C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C22×D4, C2×C4○D4, C2×C4.4D4, D4○SD16, Q8○D8, C42.367C23

Smallest permutation representation of C42.367C23
On 64 points
Generators in S64
(1 23 11 43)(2 24 12 44)(3 17 13 45)(4 18 14 46)(5 19 15 47)(6 20 16 48)(7 21 9 41)(8 22 10 42)(25 51 39 62)(26 52 40 63)(27 53 33 64)(28 54 34 57)(29 55 35 58)(30 56 36 59)(31 49 37 60)(32 50 38 61)
(1 9 5 13)(2 10 6 14)(3 11 7 15)(4 12 8 16)(17 43 21 47)(18 44 22 48)(19 45 23 41)(20 46 24 42)(25 37 29 33)(26 38 30 34)(27 39 31 35)(28 40 32 36)(49 58 53 62)(50 59 54 63)(51 60 55 64)(52 61 56 57)
(1 43 5 47)(2 18 6 22)(3 41 7 45)(4 24 8 20)(9 17 13 21)(10 48 14 44)(11 23 15 19)(12 46 16 42)(25 62 29 58)(26 54 30 50)(27 60 31 64)(28 52 32 56)(33 49 37 53)(34 63 38 59)(35 55 39 51)(36 61 40 57)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 60 15 53)(2 61 16 54)(3 62 9 55)(4 63 10 56)(5 64 11 49)(6 57 12 50)(7 58 13 51)(8 59 14 52)(17 29 41 39)(18 30 42 40)(19 31 43 33)(20 32 44 34)(21 25 45 35)(22 26 46 36)(23 27 47 37)(24 28 48 38)

G:=sub<Sym(64)| (1,23,11,43)(2,24,12,44)(3,17,13,45)(4,18,14,46)(5,19,15,47)(6,20,16,48)(7,21,9,41)(8,22,10,42)(25,51,39,62)(26,52,40,63)(27,53,33,64)(28,54,34,57)(29,55,35,58)(30,56,36,59)(31,49,37,60)(32,50,38,61), (1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(17,43,21,47)(18,44,22,48)(19,45,23,41)(20,46,24,42)(25,37,29,33)(26,38,30,34)(27,39,31,35)(28,40,32,36)(49,58,53,62)(50,59,54,63)(51,60,55,64)(52,61,56,57), (1,43,5,47)(2,18,6,22)(3,41,7,45)(4,24,8,20)(9,17,13,21)(10,48,14,44)(11,23,15,19)(12,46,16,42)(25,62,29,58)(26,54,30,50)(27,60,31,64)(28,52,32,56)(33,49,37,53)(34,63,38,59)(35,55,39,51)(36,61,40,57), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,60,15,53)(2,61,16,54)(3,62,9,55)(4,63,10,56)(5,64,11,49)(6,57,12,50)(7,58,13,51)(8,59,14,52)(17,29,41,39)(18,30,42,40)(19,31,43,33)(20,32,44,34)(21,25,45,35)(22,26,46,36)(23,27,47,37)(24,28,48,38)>;

G:=Group( (1,23,11,43)(2,24,12,44)(3,17,13,45)(4,18,14,46)(5,19,15,47)(6,20,16,48)(7,21,9,41)(8,22,10,42)(25,51,39,62)(26,52,40,63)(27,53,33,64)(28,54,34,57)(29,55,35,58)(30,56,36,59)(31,49,37,60)(32,50,38,61), (1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(17,43,21,47)(18,44,22,48)(19,45,23,41)(20,46,24,42)(25,37,29,33)(26,38,30,34)(27,39,31,35)(28,40,32,36)(49,58,53,62)(50,59,54,63)(51,60,55,64)(52,61,56,57), (1,43,5,47)(2,18,6,22)(3,41,7,45)(4,24,8,20)(9,17,13,21)(10,48,14,44)(11,23,15,19)(12,46,16,42)(25,62,29,58)(26,54,30,50)(27,60,31,64)(28,52,32,56)(33,49,37,53)(34,63,38,59)(35,55,39,51)(36,61,40,57), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,60,15,53)(2,61,16,54)(3,62,9,55)(4,63,10,56)(5,64,11,49)(6,57,12,50)(7,58,13,51)(8,59,14,52)(17,29,41,39)(18,30,42,40)(19,31,43,33)(20,32,44,34)(21,25,45,35)(22,26,46,36)(23,27,47,37)(24,28,48,38) );

G=PermutationGroup([[(1,23,11,43),(2,24,12,44),(3,17,13,45),(4,18,14,46),(5,19,15,47),(6,20,16,48),(7,21,9,41),(8,22,10,42),(25,51,39,62),(26,52,40,63),(27,53,33,64),(28,54,34,57),(29,55,35,58),(30,56,36,59),(31,49,37,60),(32,50,38,61)], [(1,9,5,13),(2,10,6,14),(3,11,7,15),(4,12,8,16),(17,43,21,47),(18,44,22,48),(19,45,23,41),(20,46,24,42),(25,37,29,33),(26,38,30,34),(27,39,31,35),(28,40,32,36),(49,58,53,62),(50,59,54,63),(51,60,55,64),(52,61,56,57)], [(1,43,5,47),(2,18,6,22),(3,41,7,45),(4,24,8,20),(9,17,13,21),(10,48,14,44),(11,23,15,19),(12,46,16,42),(25,62,29,58),(26,54,30,50),(27,60,31,64),(28,52,32,56),(33,49,37,53),(34,63,38,59),(35,55,39,51),(36,61,40,57)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,60,15,53),(2,61,16,54),(3,62,9,55),(4,63,10,56),(5,64,11,49),(6,57,12,50),(7,58,13,51),(8,59,14,52),(17,29,41,39),(18,30,42,40),(19,31,43,33),(20,32,44,34),(21,25,45,35),(22,26,46,36),(23,27,47,37),(24,28,48,38)]])

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I···4O8A8B8C8D8E···8J
order1222222444444444···488888···8
size1111228222244448···822224···4

32 irreducible representations

dim11111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○SD16Q8○D8
kernelC42.367C23C82M4(2)C2×Q8⋊C4C23.24D4C23.36D4C23.38D4C4.SD16C42.78C22C42.28C22C42.30C22C23.38C23C23.41C23C22⋊C4C4⋊C4C2×C4C2C2
# reps11111122221122822

Matrix representation of C42.367C23 in GL6(𝔽17)

1380000
040000
004000
000400
0000130
0000013
,
100000
010000
000100
0016000
000001
0000160
,
1380000
1340000
0013000
000400
0000130
000004
,
400000
040000
003300
0014300
000033
0000143
,
490000
0130000
0000130
0000013
004000
000400

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,8,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,13,0,0,0,0,8,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,14,0,0,0,0,3,3,0,0,0,0,0,0,3,14,0,0,0,0,3,3],[4,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,13,0,0,0,0,0,0,13,0,0] >;

C42.367C23 in GAP, Magma, Sage, TeX

C_4^2._{367}C_2^3
% in TeX

G:=Group("C4^2.367C2^3");
// GroupNames label

G:=SmallGroup(128,1869);
// by ID

G=gap.SmallGroup(128,1869);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,232,758,100,1018,2804,172,4037,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b^2,d^2=a^2*b^-1,e^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
// generators/relations

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