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G = C3×C42.C2order 96 = 25·3

Direct product of C3 and C42.C2

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

Aliases: C3×C42.C2, C42.4C6, C12.11Q8, C4⋊C4.4C6, C4.3(C3×Q8), C2.4(C6×Q8), C6.21(C2×Q8), (C4×C12).10C2, C6.45(C4○D4), (C2×C6).80C23, (C2×C12).67C22, C22.15(C22×C6), (C2×C4).7(C2×C6), C2.8(C3×C4○D4), (C3×C4⋊C4).11C2, SmallGroup(96,172)

Series: Derived Chief Lower central Upper central

C1C22 — C3×C42.C2
C1C2C22C2×C6C2×C12C3×C4⋊C4 — C3×C42.C2
C1C22 — C3×C42.C2
C1C2×C6 — C3×C42.C2

Generators and relations for C3×C42.C2
 G = < a,b,c,d | a3=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >

Subgroups: 68 in 56 conjugacy classes, 44 normal (12 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×6], C22, C6, C6 [×2], C2×C4, C2×C4 [×6], C12 [×2], C12 [×6], C2×C6, C42, C4⋊C4 [×6], C2×C12, C2×C12 [×6], C42.C2, C4×C12, C3×C4⋊C4 [×6], C3×C42.C2
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], Q8 [×2], C23, C2×C6 [×7], C2×Q8, C4○D4 [×2], C3×Q8 [×2], C22×C6, C42.C2, C6×Q8, C3×C4○D4 [×2], C3×C42.C2

Smallest permutation representation of C3×C42.C2
Regular action on 96 points
Generators in S96
(1 35 11)(2 36 12)(3 33 9)(4 34 10)(5 30 14)(6 31 15)(7 32 16)(8 29 13)(17 37 41)(18 38 42)(19 39 43)(20 40 44)(21 51 45)(22 52 46)(23 49 47)(24 50 48)(25 95 69)(26 96 70)(27 93 71)(28 94 72)(53 75 79)(54 76 80)(55 73 77)(56 74 78)(57 61 83)(58 62 84)(59 63 81)(60 64 82)(65 87 91)(66 88 92)(67 85 89)(68 86 90)
(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 23 39 15)(2 24 40 16)(3 21 37 13)(4 22 38 14)(5 34 52 42)(6 35 49 43)(7 36 50 44)(8 33 51 41)(9 45 17 29)(10 46 18 30)(11 47 19 31)(12 48 20 32)(25 89 64 80)(26 90 61 77)(27 91 62 78)(28 92 63 79)(53 94 66 81)(54 95 67 82)(55 96 68 83)(56 93 65 84)(57 73 70 86)(58 74 71 87)(59 75 72 88)(60 76 69 85)
(1 74 39 87)(2 88 40 75)(3 76 37 85)(4 86 38 73)(5 28 52 63)(6 64 49 25)(7 26 50 61)(8 62 51 27)(9 54 17 67)(10 68 18 55)(11 56 19 65)(12 66 20 53)(13 58 21 71)(14 72 22 59)(15 60 23 69)(16 70 24 57)(29 84 45 93)(30 94 46 81)(31 82 47 95)(32 96 48 83)(33 80 41 89)(34 90 42 77)(35 78 43 91)(36 92 44 79)

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

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

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

C3×C42.C2 is a maximal subgroup of
C42.8D6  Dic6.4Q8  C42.68D6  C42.215D6  D12.4Q8  C42.70D6  C42.216D6  C42.71D6  Dic67Q8  C42.147D6  C42.236D6  C42.148D6  D127Q8  C42.237D6  C42.150D6  C42.151D6  C42.152D6  C42.153D6  C42.154D6  C42.155D6  C42.156D6  C42.157D6  C42.158D6

42 conjugacy classes

class 1 2A2B2C3A3B4A···4F4G4H4I4J6A···6F12A···12L12M···12T
order1222334···444446···612···1212···12
size1111112···244441···12···24···4

42 irreducible representations

dim1111112222
type+++-
imageC1C2C2C3C6C6Q8C4○D4C3×Q8C3×C4○D4
kernelC3×C42.C2C4×C12C3×C4⋊C4C42.C2C42C4⋊C4C12C6C4C2
# reps11622122448

Matrix representation of C3×C42.C2 in GL4(𝔽13) generated by

9000
0900
0010
0001
,
8000
0800
0001
00120
,
0100
1000
0001
00120
,
11900
4200
0039
00910
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,8,0,0,0,0,0,12,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,12,0,0,1,0],[11,4,0,0,9,2,0,0,0,0,3,9,0,0,9,10] >;

C3×C42.C2 in GAP, Magma, Sage, TeX

C_3\times C_4^2.C_2
% in TeX

G:=Group("C3xC4^2.C2");
// GroupNames label

G:=SmallGroup(96,172);
// by ID

G=gap.SmallGroup(96,172);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,144,313,295,938,122]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b^2*c>;
// generators/relations

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