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G = C4×C3⋊C8order 96 = 25·3

Direct product of C4 and C3⋊C8

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×C3⋊C8, C122C8, C42.6S3, C6.1C42, C31(C4×C8), C6.6(C2×C8), C4.18(C4×S3), (C4×C12).6C2, (C2×C4).87D6, C12.23(C2×C4), (C2×C12).10C4, C2.1(C4×Dic3), (C2×C4).7Dic3, C22.6(C2×Dic3), (C2×C12).101C22, C2.1(C2×C3⋊C8), C42(C2×C3⋊C8), (C2×C3⋊C8).11C2, (C2×C6).24(C2×C4), SmallGroup(96,9)

Series: Derived Chief Lower central Upper central

C1C3 — C4×C3⋊C8
C1C3C6C12C2×C12C2×C3⋊C8 — C4×C3⋊C8
C3 — C4×C3⋊C8
C1C42

Generators and relations for C4×C3⋊C8
 G = < a,b,c | a4=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

3C8
3C8
3C8
3C8
3C2×C8
3C2×C8
3C4×C8

Smallest permutation representation of C4×C3⋊C8
Regular action on 96 points
Generators in S96
(1 79 21 13)(2 80 22 14)(3 73 23 15)(4 74 24 16)(5 75 17 9)(6 76 18 10)(7 77 19 11)(8 78 20 12)(25 45 49 83)(26 46 50 84)(27 47 51 85)(28 48 52 86)(29 41 53 87)(30 42 54 88)(31 43 55 81)(32 44 56 82)(33 71 61 95)(34 72 62 96)(35 65 63 89)(36 66 64 90)(37 67 57 91)(38 68 58 92)(39 69 59 93)(40 70 60 94)
(1 43 67)(2 68 44)(3 45 69)(4 70 46)(5 47 71)(6 72 48)(7 41 65)(8 66 42)(9 27 33)(10 34 28)(11 29 35)(12 36 30)(13 31 37)(14 38 32)(15 25 39)(16 40 26)(17 85 95)(18 96 86)(19 87 89)(20 90 88)(21 81 91)(22 92 82)(23 83 93)(24 94 84)(49 59 73)(50 74 60)(51 61 75)(52 76 62)(53 63 77)(54 78 64)(55 57 79)(56 80 58)
(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)

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

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

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

C4×C3⋊C8 is a maximal subgroup of
C42.279D6  C24⋊C8  C12.53D8  C12.39SD16  D122C8  Dic62C8  C12.57D8  C12.26Q16  S3×C4×C8  C42.282D6  D6.4C42  C42.185D6  C42.196D6  Dic6⋊C8  C42.198D6  D12⋊C8  C122M4(2)  C42.285D6  C12.5C42  C42.187D6  C123M4(2)  C42.210D6  C42.213D6  C42.214D6  C42.215D6  C42.216D6  C12.16D8  C12⋊D8  C124SD16  C12.17D8  C12.9Q16  C12.SD16  C126SD16  C12.D8  C123Q16  C12.Q16
C4×C3⋊C8 is a maximal quotient of
C24⋊C8  C24.C8  (C2×C12)⋊3C8

48 conjugacy classes

class 1 2A2B2C 3 4A···4L6A6B6C8A···8P12A···12L
order122234···46668···812···12
size111121···12223···32···2

48 irreducible representations

dim11111122222
type++++-+
imageC1C2C2C4C4C8S3Dic3D6C3⋊C8C4×S3
kernelC4×C3⋊C8C2×C3⋊C8C4×C12C3⋊C8C2×C12C12C42C2×C4C2×C4C4C4
# reps121841612184

Matrix representation of C4×C3⋊C8 in GL3(𝔽73) generated by

4600
0270
0027
,
100
0072
0172
,
2700
0236
02950
G:=sub<GL(3,GF(73))| [46,0,0,0,27,0,0,0,27],[1,0,0,0,0,1,0,72,72],[27,0,0,0,23,29,0,6,50] >;

C4×C3⋊C8 in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes C_8
% in TeX

G:=Group("C4xC3:C8");
// GroupNames label

G:=SmallGroup(96,9);
// by ID

G=gap.SmallGroup(96,9);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,55,86,2309]);
// Polycyclic

G:=Group<a,b,c|a^4=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×C3⋊C8 in TeX

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