C12:2M4(2) - GroupNames

G = C₁₂⋊₂M₄(2) order 192 = 2⁶·3

2^nd semidirect product of C₁₂ and M₄(2) acting via M₄(2)/C₄=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₂⋊₂M₄(2), C₄².₂₀₄D₆, C₃⋊C₈⋊₁₇D₄, C₄⋊C₈⋊₁₄S₃, D₆⋊C₈⋊₂₉C₂, C₃⋊₃(C₈⋊₆D₄), C₆.₄₈(C₄×D₄), C₄⋊₁(C₈⋊S₃), D₆⋊C₄.₁₂C₄, (C₄×D₁₂).₉C₂, (C₂×C₈).₁₈₃D₆, C₄.₂₀₈(S₃×D₄), (C₂×D₁₂).₁₃C₄, C₆.₂₉(C₈○D₄), C₁₂.₃₆₇(C₂×D₄), C₄⋊Dic₃.₁₉C₄, C₆.₉(C₂×M₄(2)), (C₄×C₁₂).₆₄C₂², C₁₂.₃₃₆(C₄○D₄), C₂.₈(Dic₃⋊₅D₄), (C₂×C₁₂).₈₃₅C₂³, C₂.₁₃(D₁₂.C₄), (C₂×C₂₄).₂₅₇C₂², C₄.₅₆(Q₈⋊₃S₃), (C₄×C₃⋊C₈)⋊₅C₂, (C₃×C₄⋊C₈)⋊₂₄C₂, (C₂×C₄).₇₃(C₄×S₃), (C₂×C₈⋊S₃)⋊₂₂C₂, C₂.₁₄(C₂×C₈⋊S₃), C₂².₁₁₃(S₃×C₂×C₄), (C₂×C₁₂).₁₅₉(C₂×C₄), (C₂×C₃⋊C₈).₃₀₇C₂², (S₃×C₂×C₄).₁₈₁C₂², (C₂×C₆).₉₀(C₂²×C₄), (C₂²×S₃).₁₆(C₂×C₄), (C₂×C₄).₇₇₇(C₂²×S₃), (C₂×Dic₃).₂₃(C₂×C₄), SmallGroup(192,397)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₁₂⋊₂M₄(2)

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — S₃×C₂×C₄ — C₄×D₁₂ — C₁₂⋊₂M₄(2)

Lower central

C₃ — C₂×C₆ — C₁₂⋊₂M₄(2)

Upper central

C₁ — C₂×C₄ — C₄⋊C₈

Generators and relations for C₁₂⋊₂M₄(2)
G = < a,b,c | a¹²=b⁸=c²=1, bab^-1=a⁵, cac=a^-1, cbc=b⁵ >

Subgroups: 312 in 122 conjugacy classes, 53 normal (31 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², S₃, C₆, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, Dic₃, C₁₂, C₁₂, C₁₂, D₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂×D₄, C₃⋊C₈, C₃⋊C₈, C₂₄, C₄×S₃, D₁₂, C₂×Dic₃, C₂×C₁₂, C₂²×S₃, C₄×C₈, C₂²⋊C₈, C₄⋊C₈, C₄×D₄, C₂×M₄(2), C₈⋊S₃, C₂×C₃⋊C₈, C₄⋊Dic₃, D₆⋊C₄, C₄×C₁₂, C₂×C₂₄, S₃×C₂×C₄, C₂×D₁₂, C₈⋊₆D₄, C₄×C₃⋊C₈, D₆⋊C₈, C₃×C₄⋊C₈, C₄×D₁₂, C₂×C₈⋊S₃, C₁₂⋊₂M₄(2)
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, C₂³, D₆, M₄(2), C₂²×C₄, C₂×D₄, C₄○D₄, C₄×S₃, C₂²×S₃, C₄×D₄, C₂×M₄(2), C₈○D₄, C₈⋊S₃, S₃×C₂×C₄, S₃×D₄, Q₈⋊₃S₃, C₈⋊₆D₄, Dic₃⋊₅D₄, C₂×C₈⋊S₃, D₁₂.C₄, C₁₂⋊₂M₄(2)

Smallest permutation representation of C₁₂⋊₂M₄(2)
►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 60 30 69 75 13 94 45)(2 53 31 62 76 18 95 38)(3 58 32 67 77 23 96 43)(4 51 33 72 78 16 85 48)(5 56 34 65 79 21 86 41)(6 49 35 70 80 14 87 46)(7 54 36 63 81 19 88 39)(8 59 25 68 82 24 89 44)(9 52 26 61 83 17 90 37)(10 57 27 66 84 22 91 42)(11 50 28 71 73 15 92 47)(12 55 29 64 74 20 93 40)

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+				+	+	+	+						+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	S₃	D₄	D₆	D₆	M₄(2)	C₄○D₄	C₄×S₃	C₈○D₄	C₈⋊S₃	S₃×D₄	Q₈⋊₃S₃	D₁₂.C₄
kernel	C₁₂⋊₂M₄(2)	C₄×C₃⋊C₈	D₆⋊C₈	C₃×C₄⋊C₈	C₄×D₁₂	C₂×C₈⋊S₃	C₄⋊Dic₃	D₆⋊C₄	C₂×D₁₂	C₄⋊C₈	C₃⋊C₈	C₄²	C₂×C₈	C₁₂	C₁₂	C₂×C₄	C₆	C₄	C₄	C₄	C₂
# reps	1	1	2	1	1	2	2	4	2	1	2	1	2	4	2	4	4	8	1	1	2

Matrix representation of C₁₂⋊₂M₄(2) ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	72	0	0
0	0	1	1	0	0
0	0	0	0	27	71
0	0	0	0	0	46

49	3	0	0	0	0
18	24	0	0	0	0
0	0	72	0	0	0
0	0	1	1	0	0
0	0	0	0	27	0
0	0	0	0	0	27

1	0	0	0	0	0
16	72	0	0	0	0
0	0	1	0	0	0
0	0	72	72	0	0
0	0	0	0	1	0
0	0	0	0	27	72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,1,0,0,0,0,0,0,27,0,0,0,0,0,71,46],[49,18,0,0,0,0,3,24,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[1,16,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,27,0,0,0,0,0,72] >;

C₁₂⋊₂M₄(2) in GAP, Magma, Sage, TeX

C_{12}\rtimes_2M_4(2)

% in TeX

G:=Group("C12:2M4(2)");

// GroupNames label

G:=SmallGroup(192,397);

// by ID

G=gap.SmallGroup(192,397);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,758,219,58,136,6278]);

// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^5>;

// generators/relations

G = C12⋊2M4(2) order 192 = 26·3

2nd semidirect product of C12 and M4(2) acting via M4(2)/C4=C22

G = C₁₂⋊₂M₄(2) order 192 = 2⁶·3

2^nd semidirect product of C₁₂ and M₄(2) acting via M₄(2)/C₄=C₂²