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G = C6.C42order 96 = 25·3

5th non-split extension by C6 of C42 acting via C42/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.5C42, C23.32D6, C22.11D12, C22.3Dic6, (C2×C12)⋊3C4, C6.9(C4⋊C4), (C2×C6).4Q8, (C2×C4)⋊2Dic3, (C2×C6).32D4, C2.2(D6⋊C4), (C2×Dic3)⋊2C4, C3⋊(C2.C42), (C22×C4).4S3, C2.5(C4×Dic3), (C22×C12).1C2, C22.12(C4×S3), C2.2(C4⋊Dic3), C6.11(C22⋊C4), C2.2(Dic3⋊C4), C2.2(C6.D4), C22.16(C3⋊D4), (C22×C6).31C22, (C22×Dic3).1C2, C22.10(C2×Dic3), (C2×C6).13(C2×C4), SmallGroup(96,38)

Series: Derived Chief Lower central Upper central

C1C6 — C6.C42
C1C3C6C2×C6C22×C6C22×Dic3 — C6.C42
C3C6 — C6.C42
C1C23C22×C4

Generators and relations for C6.C42
 G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=abc, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd5 >

Subgroups: 146 in 76 conjugacy classes, 45 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×10], C23, Dic3 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×4], C22×C4, C22×C4 [×2], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×2], C22×C6, C2.C42, C22×Dic3 [×2], C22×C12, C6.C42
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], D4 [×3], Q8, Dic3 [×2], D6, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic6, C4×S3 [×2], D12, C2×Dic3, C3⋊D4 [×2], C2.C42, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, C6.C42

Smallest permutation representation of C6.C42
Regular action on 96 points
Generators in S96
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 49)(23 50)(24 51)(25 68)(26 69)(27 70)(28 71)(29 72)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 85)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 82)(14 83)(15 84)(16 73)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 69)(38 70)(39 71)(40 72)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)
(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 25 16 42)(2 90 17 67)(3 35 18 40)(4 88 19 65)(5 33 20 38)(6 86 21 63)(7 31 22 48)(8 96 23 61)(9 29 24 46)(10 94 13 71)(11 27 14 44)(12 92 15 69)(26 78 43 54)(28 76 45 52)(30 74 47 50)(32 84 37 60)(34 82 39 58)(36 80 41 56)(49 85 73 62)(51 95 75 72)(53 93 77 70)(55 91 79 68)(57 89 81 66)(59 87 83 64)

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

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

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

C6.C42 is a maximal subgroup of
C23.35D12  C22.2D24  (C6×D4)⋊C4  (C6×Q8)⋊C4  (C2×C12)⋊Q8  C6.(C4×Q8)  Dic3.5C42  Dic3⋊C42  C3⋊(C428C4)  C3⋊(C425C4)  C6.(C4×D4)  C2.(C4×D12)  C2.(C4×Dic6)  Dic3⋊C4⋊C4  (C2×C4)⋊Dic6  C6.(C4⋊Q8)  (C2×Dic3).9D4  (C2×C4).17D12  (C2×C4).Dic6  (C22×C4).85D6  (C22×C4).30D6  S3×C2.C42  C22.58(S3×D4)  (C2×C4)⋊9D12  D6⋊C42  D6⋊(C4⋊C4)  D6⋊C4⋊C4  D6⋊C45C4  D6⋊C43C4  (C22×S3)⋊Q8  (C2×C4).21D12  C6.(C4⋊D4)  C124(C4⋊C4)  (C2×Dic6)⋊7C4  C4×Dic3⋊C4  C426Dic3  (C2×C42).6S3  C4×C4⋊Dic3  C4211Dic3  C427Dic3  C4×D6⋊C4  (C2×C42)⋊3S3  Dic3×C22⋊C4  C24.55D6  C24.56D6  C24.14D6  C24.15D6  C24.57D6  C232Dic6  C24.17D6  C24.18D6  C24.58D6  C24.19D6  C24.20D6  C24.21D6  C24.23D6  C24.60D6  C24.25D6  C233D12  C24.27D6  C12⋊(C4⋊C4)  C4.(D6⋊C4)  Dic3×C4⋊C4  Dic3⋊(C4⋊C4)  (C4×Dic3)⋊9C4  C6.67(C4×D4)  (C2×Dic3)⋊Q8  C4⋊C45Dic3  (C2×C4).44D12  (C2×C12).54D4  (C2×Dic3).Q8  C4⋊C46Dic3  (C2×C12).288D4  (C2×C12).55D4  C4⋊(D6⋊C4)  D6⋊C46C4  D6⋊C47C4  (C2×C4)⋊3D12  (C2×C12).289D4  (C2×C12).290D4  (C2×C12).56D4  C4×C6.D4  C24.73D6  C24.74D6  C24.75D6  C24.76D6  C24.29D6  C24.31D6  C24.32D6  (C6×Q8)⋊7C4  C22.52(S3×Q8)  (C22×Q8)⋊9S3  C18.C42  C62.6Q8  C62.15Q8  C30.24C42  C30.29C42  D10.20D12  D10.10D12
C6.C42 is a maximal quotient of
C12.8C42  (C2×C12)⋊3C8  C24.12D6  C24.13D6  C12.C42  C12.(C4⋊C4)  C423Dic3  C12.2C42  (C2×C12).Q8  (C2×C24)⋊5C4  C12.9C42  C12.10C42  M4(2)⋊Dic3  C12.3C42  (C2×C24)⋊C4  C12.20C42  C12.4C42  M4(2)⋊4Dic3  C12.21C42  C18.C42  C62.6Q8  C62.15Q8  C30.24C42  C30.29C42  D10.20D12  D10.10D12

36 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E···4L6A···6G12A···12H
order12···2344444···46···612···12
size11···1222226···62···22···2

36 irreducible representations

dim11111222222222
type+++++--+-+
imageC1C2C2C4C4S3D4Q8Dic3D6Dic6C4×S3D12C3⋊D4
kernelC6.C42C22×Dic3C22×C12C2×Dic3C2×C12C22×C4C2×C6C2×C6C2×C4C23C22C22C22C22
# reps12184131212424

Matrix representation of C6.C42 in GL4(𝔽13) generated by

12000
0100
0010
0001
,
1000
01200
00120
00012
,
1000
0100
00120
00012
,
1000
0500
00103
00107
,
8000
0800
00103
0063
G:=sub<GL(4,GF(13))| [12,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,5,0,0,0,0,10,10,0,0,3,7],[8,0,0,0,0,8,0,0,0,0,10,6,0,0,3,3] >;

C6.C42 in GAP, Magma, Sage, TeX

C_6.C_4^2
% in TeX

G:=Group("C6.C4^2");
// GroupNames label

G:=SmallGroup(96,38);
// by ID

G=gap.SmallGroup(96,38);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,217,55,2309]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=b,e^2=a*b*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^5>;
// generators/relations

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