C24:C2:C4 - GroupNames

G = C₂₄⋊C₂⋊C₄ order 192 = 2⁶·3

3^rd semidirect product of C₂₄⋊C₂ and C₄ acting via C₄/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₈⋊₆(C₄×S₃), C₂₄⋊C₂⋊₃C₄, C₂₄⋊₁₁(C₂×C₄), C₂₄⋊C₄⋊₇C₂, (C₂×C₈).₆₇D₆, C₆.₅₄(C₄×D₄), C₂.D₈⋊₁₂S₃, C₄⋊C₄.₁₇₂D₆, Dic₆⋊₇(C₂×C₄), D₁₂.₉(C₂×C₄), C₂.₆(D₈⋊S₃), C₆.D₈.₇C₂, C₂².₉₂(S₃×D₄), Dic₆⋊C₄⋊₇C₂, Dic₃⋊₅D₄.₇C₂, C₁₂.₄₄(C₄○D₄), C₆.SD₁₆⋊₂₃C₂, C₃⋊₄(SD₁₆⋊C₄), C₆.₄₄(C₈⋊C₂²), C₁₂.₅₁(C₂²×C₄), C₂.₆(Q₁₆⋊S₃), (C₂×C₂₄).₁₄₅C₂², (C₂×C₁₂).₃₀₅C₂³, C₄.₁₂(Q₈⋊₃S₃), (C₂×Dic₃).₁₇₀D₄, (C₂×D₁₂).₈₅C₂², C₆.₇₃(C₈.C₂²), C₂.₁₄(Dic₃⋊₅D₄), (C₄×Dic₃).₃₇C₂², (C₂×Dic₆).₉₃C₂², C₄.₄₅(S₃×C₂×C₄), (C₃×C₂.D₈)⋊₉C₂, (C₂×C₂₄⋊C₂).₇C₂, (C₂×C₆).₃₁₀(C₂×D₄), (C₂×C₃⋊C₈).₇₄C₂², (C₃×C₄⋊C₄).₉₈C₂², (C₂×C₄).₄₀₈(C₂²×S₃), SmallGroup(192,448)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₂₄⋊C₂⋊C₄

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — C₂×D₁₂ — C₂×C₂₄⋊C₂ — C₂₄⋊C₂⋊C₄

Lower central

C₃ — C₆ — C₁₂ — C₂₄⋊C₂⋊C₄

Upper central

C₁ — C₂² — C₂×C₄ — C₂.D₈

Generators and relations for C₂₄⋊C₂⋊C₄
G = < a,b,c | a²⁴=b²=c⁴=1, bab=a¹¹, cac^-1=a⁷, bc=cb >

Subgroups: 352 in 120 conjugacy classes, 49 normal (37 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₃⋊C₈, C₂₄, Dic₆, Dic₆, C₄×S₃, D₁₂, D₁₂, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂²×S₃, C₈⋊C₄, D₄⋊C₄, Q₈⋊C₄, C₂.D₈, C₄×D₄, C₄×Q₈, C₂×SD₁₆, C₂₄⋊C₂, C₂×C₃⋊C₈, C₄×Dic₃, C₄×Dic₃, Dic₃⋊C₄, D₆⋊C₄, C₃×C₄⋊C₄, C₂×C₂₄, C₂×Dic₆, S₃×C₂×C₄, C₂×D₁₂, SD₁₆⋊C₄, C₆.D₈, C₆.SD₁₆, C₂₄⋊C₄, C₃×C₂.D₈, Dic₆⋊C₄, Dic₃⋊₅D₄, C₂×C₂₄⋊C₂, C₂₄⋊C₂⋊C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, C₂³, D₆, C₂²×C₄, C₂×D₄, C₄○D₄, C₄×S₃, C₂²×S₃, C₄×D₄, C₈⋊C₂², C₈.C₂², S₃×C₂×C₄, S₃×D₄, Q₈⋊₃S₃, SD₁₆⋊C₄, Dic₃⋊₅D₄, D₈⋊S₃, Q₁₆⋊S₃, C₂₄⋊C₂⋊C₄

Smallest permutation representation of C₂₄⋊C₂⋊C₄
►On 96 points

Generators in S₉₆

(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 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

(1 66)(2 53)(3 64)(4 51)(5 62)(6 49)(7 60)(8 71)(9 58)(10 69)(11 56)(12 67)(13 54)(14 65)(15 52)(16 63)(17 50)(18 61)(19 72)(20 59)(21 70)(22 57)(23 68)(24 55)(25 75)(26 86)(27 73)(28 84)(29 95)(30 82)(31 93)(32 80)(33 91)(34 78)(35 89)(36 76)(37 87)(38 74)(39 85)(40 96)(41 83)(42 94)(43 81)(44 92)(45 79)(46 90)(47 77)(48 88)

(1 77 66 47)(2 84 67 30)(3 91 68 37)(4 74 69 44)(5 81 70 27)(6 88 71 34)(7 95 72 41)(8 78 49 48)(9 85 50 31)(10 92 51 38)(11 75 52 45)(12 82 53 28)(13 89 54 35)(14 96 55 42)(15 79 56 25)(16 86 57 32)(17 93 58 39)(18 76 59 46)(19 83 60 29)(20 90 61 36)(21 73 62 43)(22 80 63 26)(23 87 64 33)(24 94 65 40)

G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,66)(2,53)(3,64)(4,51)(5,62)(6,49)(7,60)(8,71)(9,58)(10,69)(11,56)(12,67)(13,54)(14,65)(15,52)(16,63)(17,50)(18,61)(19,72)(20,59)(21,70)(22,57)(23,68)(24,55)(25,75)(26,86)(27,73)(28,84)(29,95)(30,82)(31,93)(32,80)(33,91)(34,78)(35,89)(36,76)(37,87)(38,74)(39,85)(40,96)(41,83)(42,94)(43,81)(44,92)(45,79)(46,90)(47,77)(48,88), (1,77,66,47)(2,84,67,30)(3,91,68,37)(4,74,69,44)(5,81,70,27)(6,88,71,34)(7,95,72,41)(8,78,49,48)(9,85,50,31)(10,92,51,38)(11,75,52,45)(12,82,53,28)(13,89,54,35)(14,96,55,42)(15,79,56,25)(16,86,57,32)(17,93,58,39)(18,76,59,46)(19,83,60,29)(20,90,61,36)(21,73,62,43)(22,80,63,26)(23,87,64,33)(24,94,65,40)>;

G:=Group( (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,66)(2,53)(3,64)(4,51)(5,62)(6,49)(7,60)(8,71)(9,58)(10,69)(11,56)(12,67)(13,54)(14,65)(15,52)(16,63)(17,50)(18,61)(19,72)(20,59)(21,70)(22,57)(23,68)(24,55)(25,75)(26,86)(27,73)(28,84)(29,95)(30,82)(31,93)(32,80)(33,91)(34,78)(35,89)(36,76)(37,87)(38,74)(39,85)(40,96)(41,83)(42,94)(43,81)(44,92)(45,79)(46,90)(47,77)(48,88), (1,77,66,47)(2,84,67,30)(3,91,68,37)(4,74,69,44)(5,81,70,27)(6,88,71,34)(7,95,72,41)(8,78,49,48)(9,85,50,31)(10,92,51,38)(11,75,52,45)(12,82,53,28)(13,89,54,35)(14,96,55,42)(15,79,56,25)(16,86,57,32)(17,93,58,39)(18,76,59,46)(19,83,60,29)(20,90,61,36)(21,73,62,43)(22,80,63,26)(23,87,64,33)(24,94,65,40) );

G=PermutationGroup([[(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,66),(2,53),(3,64),(4,51),(5,62),(6,49),(7,60),(8,71),(9,58),(10,69),(11,56),(12,67),(13,54),(14,65),(15,52),(16,63),(17,50),(18,61),(19,72),(20,59),(21,70),(22,57),(23,68),(24,55),(25,75),(26,86),(27,73),(28,84),(29,95),(30,82),(31,93),(32,80),(33,91),(34,78),(35,89),(36,76),(37,87),(38,74),(39,85),(40,96),(41,83),(42,94),(43,81),(44,92),(45,79),(46,90),(47,77),(48,88)], [(1,77,66,47),(2,84,67,30),(3,91,68,37),(4,74,69,44),(5,81,70,27),(6,88,71,34),(7,95,72,41),(8,78,49,48),(9,85,50,31),(10,92,51,38),(11,75,52,45),(12,82,53,28),(13,89,54,35),(14,96,55,42),(15,79,56,25),(16,86,57,32),(17,93,58,39),(18,76,59,46),(19,83,60,29),(20,90,61,36),(21,73,62,43),(22,80,63,26),(23,87,64,33),(24,94,65,40)]])

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	6A	6B	6C	8A	8B	8C	8D	12A	12B	12C	12D	12E	12F	24A	24B	24C	24D
order	1	2	2	2	2	2	3	4	4	4	4	4	4	4	4	4	4	4	4	6	6	6	8	8	8	8	12	12	12	12	12	12	24	24	24	24
size	1	1	1	1	12	12	2	2	2	4	4	4	4	6	6	6	6	12	12	2	2	2	4	4	12	12	4	4	8	8	8	8	4	4	4	4

36 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+		+	+	+	+			+	-	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	S₃	D₄	D₆	D₆	C₄○D₄	C₄×S₃	C₈⋊C₂²	C₈.C₂²	Q₈⋊₃S₃	S₃×D₄	D₈⋊S₃	Q₁₆⋊S₃
kernel	C₂₄⋊C₂⋊C₄	C₆.D₈	C₆.SD₁₆	C₂₄⋊C₄	C₃×C₂.D₈	Dic₆⋊C₄	Dic₃⋊₅D₄	C₂×C₂₄⋊C₂	C₂₄⋊C₂	C₂.D₈	C₂×Dic₃	C₄⋊C₄	C₂×C₈	C₁₂	C₈	C₆	C₆	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	8	1	2	2	1	2	4	1	1	1	1	2	2

Matrix representation of C₂₄⋊C₂⋊C₄ ►in GL₆(𝔽₇₃)

0	72	0	0	0	0
1	1	0	0	0	0
0	0	11	42	11	42
0	0	31	42	31	42
0	0	62	31	11	42
0	0	42	31	31	42

72	0	0	0	0	0
1	1	0	0	0	0
0	0	1	0	0	0
0	0	72	72	0	0
0	0	0	0	72	0
0	0	0	0	1	1

46	0	0	0	0	0
0	46	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	72	0
0	0	0	0	0	72

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,11,31,62,42,0,0,42,42,31,31,0,0,11,31,11,31,0,0,42,42,42,42],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

C₂₄⋊C₂⋊C₄ in GAP, Magma, Sage, TeX

C_{24}\rtimes C_2\rtimes C_4

% in TeX

G:=Group("C24:C2:C4");

// GroupNames label

G:=SmallGroup(192,448);

// by ID

G=gap.SmallGroup(192,448);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,219,58,1684,438,102,6278]);

// Polycyclic

G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^11,c*a*c^-1=a^7,b*c=c*b>;

// generators/relations

G = C24⋊C2⋊C4 order 192 = 26·3

3rd semidirect product of C24⋊C2 and C4 acting via C4/C2=C2

G = C₂₄⋊C₂⋊C₄ order 192 = 2⁶·3

3^rd semidirect product of C₂₄⋊C₂ and C₄ acting via C₄/C₂=C₂