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G = C12⋊7M4(2)  order 192 = 26·3

1st semidirect product of C12 and M4(2) acting via M4(2)/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12⋊7M4(2), C42.269D6, C42.12Dic3, C12⋊C8⋊4C2, (C4×C12).11C4, C4.80(C2×D12), (C2×C4).87D12, C12.31(C4⋊C4), (C2×C12).59Q8, C12.83(C2×Q8), (C2×C12).396D4, C12.300(C2×D4), (C2×C42).11S3, C4⋊2(C4.Dic3), (C2×C4).44Dic6, C4.48(C2×Dic6), C3⋊2(C4⋊M4(2)), (C22×C12).23C4, C4.14(C4⋊Dic3), (C22×C4).432D6, C6.37(C2×M4(2)), (C2×C12).842C23, (C4×C12).330C22, (C22×C4).18Dic3, C23.30(C2×Dic3), C22.12(C4⋊Dic3), (C22×C12).535C22, C22.34(C22×Dic3), C6.23(C2×C4⋊C4), (C2×C4×C12).18C2, C2.4(C2×C4⋊Dic3), (C2×C6).43(C4⋊C4), (C2×C12).295(C2×C4), C2.6(C2×C4.Dic3), (C2×C3⋊C8).198C22, (C2×C4).58(C2×Dic3), (C2×C4.Dic3).3C2, (C22×C6).130(C2×C4), (C2×C4).784(C22×S3), (C2×C6).171(C22×C4), SmallGroup(192,483)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C12⋊7M4(2)
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C12⋊C8 — C12⋊7M4(2)
Lower central
C3 — C2×C6 — C12⋊7M4(2)
Upper central
C1 — C2×C4 — C2×C42

Generators and relations for C12⋊7M4(2)
 G = < a,b,c | a12=b8=c2=1, bab-1=a-1, ac=ca, cbc=b5 >

Subgroups: 216 in 126 conjugacy classes, 79 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C3⋊C8, C2×C12, C2×C12, C2×C12, C22×C6, C4⋊C8, C2×C42, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4×C12, C4×C12, C22×C12, C22×C12, C4⋊M4(2), C12⋊C8, C2×C4.Dic3, C2×C4×C12, C12⋊7M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, Dic6, D12, C2×Dic3, C22×S3, C2×C4⋊C4, C2×M4(2), C4.Dic3, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, C4⋊M4(2), C2×C4.Dic3, C2×C4⋊Dic3, C12⋊7M4(2)

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E···4N6A···6G8A···8H12A···12X
order122222344444···46···68···812···12
size111122211112···22···212···122···2

60 irreducible representations

dim11111122222222222
type++++++--+-+-+
imageC1C2C2C2C4C4S3D4Q8Dic3D6Dic3D6M4(2)Dic6D12C4.Dic3
kernelC12⋊7M4(2)C12⋊C8C2×C4.Dic3C2×C4×C12C4×C12C22×C12C2×C42C2×C12C2×C12C42C42C22×C4C22×C4C12C2×C4C2×C4C4
# reps142144122222184416

Matrix representation of C12⋊7M4(2) ►in GL4(𝔽73) generated by

9000
06500
00722
00721
,
0100
27000
00061
00670
,
1000
07200
0010
0001
G:=sub<GL(4,GF(73))| [9,0,0,0,0,65,0,0,0,0,72,72,0,0,2,1],[0,27,0,0,1,0,0,0,0,0,0,67,0,0,61,0],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1] >;
 

C12⋊7M4(2) in GAP, Magma, Sage, TeX 

C_{12}\rtimes_7M_4(2)
 
% in TeX
 
G:=Group("C12:7M4(2)");
 
// GroupNames label
 
G:=SmallGroup(192,483);
 
// by ID
 
G=gap.SmallGroup(192,483);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,253,120,758,136,6278]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^5>;
 
// generators/relations
 

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