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G = C12×M4(2)  order 192 = 26·3

Direct product of C12 and M4(2)

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C12×M4(2), C42.9C12, C12.30C42, (C4×C8)⋊13C6, C8⋊6(C2×C12), (C4×C24)⋊29C2, C24⋊28(C2×C4), C4.4(C4×C12), C8⋊C4⋊12C6, (C4×C12).24C4, (C2×C42).15C6, C22.4(C4×C12), C42.58(C2×C6), (C2×C6).11C42, C6.34(C2×C42), C2.2(C6×M4(2)), (C22×C4).13C12, C23.32(C2×C12), C4.33(C22×C12), (C22×C12).18C4, C6.46(C2×M4(2)), (C2×C12).979C23, C12.191(C22×C4), (C2×C24).445C22, (C4×C12).299C22, (C6×M4(2)).35C2, (C2×M4(2)).16C6, C22.17(C22×C12), (C22×C12).578C22, C2.6(C2×C4×C12), (C2×C4×C12).35C2, (C2×C8).99(C2×C6), (C3×C8⋊C4)⋊26C2, (C2×C4).45(C2×C12), (C2×C12).347(C2×C4), (C22×C6).113(C2×C4), (C22×C4).115(C2×C6), (C2×C4).147(C22×C6), (C2×C6).229(C22×C4), SmallGroup(192,837)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C12×M4(2)
Chief series
C1 — C2 — C22 — C2×C4 — C2×C12 — C2×C24 — C4×C24 — C12×M4(2)
Lower central
C1 — C2 — C12×M4(2)
Upper central
C1 — C4×C12 — C12×M4(2)

Generators and relations for C12×M4(2)
 G = < a,b,c | a12=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Subgroups: 162 in 142 conjugacy classes, 122 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C4×C12, C4×C12, C2×C24, C3×M4(2), C22×C12, C22×C12, C4×M4(2), C4×C24, C3×C8⋊C4, C2×C4×C12, C6×M4(2), C12×M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C42, M4(2), C22×C4, C2×C12, C22×C6, C2×C42, C2×M4(2), C4×C12, C3×M4(2), C22×C12, C4×M4(2), C2×C4×C12, C6×M4(2), C12×M4(2)

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

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

120 conjugacy classes

class 1 2A2B2C2D2E3A3B4A···4L4M···4R6A···6F6G6H6I6J8A···8P12A···12X12Y···12AJ24A···24AF
order122222334···44···46···666668···812···1212···1224···24
size111122111···12···21···122222···21···12···22···2

120 irreducible representations

dim111111111111111122
type+++++
imageC1C2C2C2C2C3C4C4C4C6C6C6C6C12C12C12M4(2)C3×M4(2)
kernelC12×M4(2)C4×C24C3×C8⋊C4C2×C4×C12C6×M4(2)C4×M4(2)C4×C12C3×M4(2)C22×C12C4×C8C8⋊C4C2×C42C2×M4(2)C42M4(2)C22×C4C12C4
# reps122122416444248328816

Matrix representation of C12×M4(2) ►in GL3(𝔽73) generated by

7000
0270
0027
,
7200
0046
010
,
7200
010
0072
G:=sub<GL(3,GF(73))| [70,0,0,0,27,0,0,0,27],[72,0,0,0,0,1,0,46,0],[72,0,0,0,1,0,0,0,72] >;
 

C12×M4(2) in GAP, Magma, Sage, TeX 

C_{12}\times M_4(2)
 
% in TeX
 
G:=Group("C12xM4(2)");
 
// GroupNames label
 
G:=SmallGroup(192,837);
 
// by ID
 
G=gap.SmallGroup(192,837);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,344,2102,172]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
 
// generators/relations
 

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