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## G = C2×C4×He3⋊C2order 432 = 24·33

### Direct product of C2×C4 and He3⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C2×C4×He3⋊C2
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×He3⋊C2 — C22×He3⋊C2 — C2×C4×He3⋊C2
 Lower central He3 — C2×C4×He3⋊C2
 Upper central C1 — C2×C12

Generators and relations for C2×C4×He3⋊C2
G = < a,b,c,d,e,f | a2=b4=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 969 in 297 conjugacy classes, 75 normal (15 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C22×C4, C3×S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, He3, C3×Dic3, C3×C12, S3×C6, C62, S3×C2×C4, C22×C12, He3⋊C2, C2×He3, C2×He3, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, He33C4, C4×He3, C2×He3⋊C2, C22×He3, S3×C2×C12, C4×He3⋊C2, C2×He33C4, C2×C4×He3, C22×He3⋊C2, C2×C4×He3⋊C2
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C3⋊S3, C4×S3, C22×S3, C2×C3⋊S3, S3×C2×C4, He3⋊C2, C4×C3⋊S3, C22×C3⋊S3, C2×He3⋊C2, C2×C4×C3⋊S3, C4×He3⋊C2, C22×He3⋊C2, C2×C4×He3⋊C2

Smallest permutation representation of C2×C4×He3⋊C2
On 72 points
Generators in S72
(1 18)(2 19)(3 20)(4 17)(5 68)(6 65)(7 66)(8 67)(9 44)(10 41)(11 42)(12 43)(13 62)(14 63)(15 64)(16 61)(21 47)(22 48)(23 45)(24 46)(25 60)(26 57)(27 58)(28 59)(29 55)(30 56)(31 53)(32 54)(33 50)(34 51)(35 52)(36 49)(37 70)(38 71)(39 72)(40 69)
(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)
(1 16 33)(2 13 34)(3 14 35)(4 15 36)(5 44 29)(6 41 30)(7 42 31)(8 43 32)(9 55 68)(10 56 65)(11 53 66)(12 54 67)(17 64 49)(18 61 50)(19 62 51)(20 63 52)(21 58 40)(22 59 37)(23 60 38)(24 57 39)(25 71 45)(26 72 46)(27 69 47)(28 70 48)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 16 58)(6 13 59)(7 14 60)(8 15 57)(9 50 69)(10 51 70)(11 52 71)(12 49 72)(17 46 54)(18 47 55)(19 48 56)(20 45 53)(25 66 63)(26 67 64)(27 68 61)(28 65 62)(33 40 44)(34 37 41)(35 38 42)(36 39 43)
(1 40 58)(2 37 59)(3 38 60)(4 39 57)(5 21 44)(6 22 41)(7 23 42)(8 24 43)(9 68 47)(10 65 48)(11 66 45)(12 67 46)(13 30 34)(14 31 35)(15 32 36)(16 29 33)(17 72 26)(18 69 27)(19 70 28)(20 71 25)(49 64 54)(50 61 55)(51 62 56)(52 63 53)
(1 20)(2 17)(3 18)(4 19)(5 11)(6 12)(7 9)(8 10)(13 49)(14 50)(15 51)(16 52)(21 45)(22 46)(23 47)(24 48)(25 40)(26 37)(27 38)(28 39)(29 53)(30 54)(31 55)(32 56)(33 63)(34 64)(35 61)(36 62)(41 67)(42 68)(43 65)(44 66)(57 70)(58 71)(59 72)(60 69)

G:=sub<Sym(72)| (1,18)(2,19)(3,20)(4,17)(5,68)(6,65)(7,66)(8,67)(9,44)(10,41)(11,42)(12,43)(13,62)(14,63)(15,64)(16,61)(21,47)(22,48)(23,45)(24,46)(25,60)(26,57)(27,58)(28,59)(29,55)(30,56)(31,53)(32,54)(33,50)(34,51)(35,52)(36,49)(37,70)(38,71)(39,72)(40,69), (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), (1,16,33)(2,13,34)(3,14,35)(4,15,36)(5,44,29)(6,41,30)(7,42,31)(8,43,32)(9,55,68)(10,56,65)(11,53,66)(12,54,67)(17,64,49)(18,61,50)(19,62,51)(20,63,52)(21,58,40)(22,59,37)(23,60,38)(24,57,39)(25,71,45)(26,72,46)(27,69,47)(28,70,48), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,16,58)(6,13,59)(7,14,60)(8,15,57)(9,50,69)(10,51,70)(11,52,71)(12,49,72)(17,46,54)(18,47,55)(19,48,56)(20,45,53)(25,66,63)(26,67,64)(27,68,61)(28,65,62)(33,40,44)(34,37,41)(35,38,42)(36,39,43), (1,40,58)(2,37,59)(3,38,60)(4,39,57)(5,21,44)(6,22,41)(7,23,42)(8,24,43)(9,68,47)(10,65,48)(11,66,45)(12,67,46)(13,30,34)(14,31,35)(15,32,36)(16,29,33)(17,72,26)(18,69,27)(19,70,28)(20,71,25)(49,64,54)(50,61,55)(51,62,56)(52,63,53), (1,20)(2,17)(3,18)(4,19)(5,11)(6,12)(7,9)(8,10)(13,49)(14,50)(15,51)(16,52)(21,45)(22,46)(23,47)(24,48)(25,40)(26,37)(27,38)(28,39)(29,53)(30,54)(31,55)(32,56)(33,63)(34,64)(35,61)(36,62)(41,67)(42,68)(43,65)(44,66)(57,70)(58,71)(59,72)(60,69)>;

G:=Group( (1,18)(2,19)(3,20)(4,17)(5,68)(6,65)(7,66)(8,67)(9,44)(10,41)(11,42)(12,43)(13,62)(14,63)(15,64)(16,61)(21,47)(22,48)(23,45)(24,46)(25,60)(26,57)(27,58)(28,59)(29,55)(30,56)(31,53)(32,54)(33,50)(34,51)(35,52)(36,49)(37,70)(38,71)(39,72)(40,69), (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), (1,16,33)(2,13,34)(3,14,35)(4,15,36)(5,44,29)(6,41,30)(7,42,31)(8,43,32)(9,55,68)(10,56,65)(11,53,66)(12,54,67)(17,64,49)(18,61,50)(19,62,51)(20,63,52)(21,58,40)(22,59,37)(23,60,38)(24,57,39)(25,71,45)(26,72,46)(27,69,47)(28,70,48), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,16,58)(6,13,59)(7,14,60)(8,15,57)(9,50,69)(10,51,70)(11,52,71)(12,49,72)(17,46,54)(18,47,55)(19,48,56)(20,45,53)(25,66,63)(26,67,64)(27,68,61)(28,65,62)(33,40,44)(34,37,41)(35,38,42)(36,39,43), (1,40,58)(2,37,59)(3,38,60)(4,39,57)(5,21,44)(6,22,41)(7,23,42)(8,24,43)(9,68,47)(10,65,48)(11,66,45)(12,67,46)(13,30,34)(14,31,35)(15,32,36)(16,29,33)(17,72,26)(18,69,27)(19,70,28)(20,71,25)(49,64,54)(50,61,55)(51,62,56)(52,63,53), (1,20)(2,17)(3,18)(4,19)(5,11)(6,12)(7,9)(8,10)(13,49)(14,50)(15,51)(16,52)(21,45)(22,46)(23,47)(24,48)(25,40)(26,37)(27,38)(28,39)(29,53)(30,54)(31,55)(32,56)(33,63)(34,64)(35,61)(36,62)(41,67)(42,68)(43,65)(44,66)(57,70)(58,71)(59,72)(60,69) );

G=PermutationGroup([[(1,18),(2,19),(3,20),(4,17),(5,68),(6,65),(7,66),(8,67),(9,44),(10,41),(11,42),(12,43),(13,62),(14,63),(15,64),(16,61),(21,47),(22,48),(23,45),(24,46),(25,60),(26,57),(27,58),(28,59),(29,55),(30,56),(31,53),(32,54),(33,50),(34,51),(35,52),(36,49),(37,70),(38,71),(39,72),(40,69)], [(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)], [(1,16,33),(2,13,34),(3,14,35),(4,15,36),(5,44,29),(6,41,30),(7,42,31),(8,43,32),(9,55,68),(10,56,65),(11,53,66),(12,54,67),(17,64,49),(18,61,50),(19,62,51),(20,63,52),(21,58,40),(22,59,37),(23,60,38),(24,57,39),(25,71,45),(26,72,46),(27,69,47),(28,70,48)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,16,58),(6,13,59),(7,14,60),(8,15,57),(9,50,69),(10,51,70),(11,52,71),(12,49,72),(17,46,54),(18,47,55),(19,48,56),(20,45,53),(25,66,63),(26,67,64),(27,68,61),(28,65,62),(33,40,44),(34,37,41),(35,38,42),(36,39,43)], [(1,40,58),(2,37,59),(3,38,60),(4,39,57),(5,21,44),(6,22,41),(7,23,42),(8,24,43),(9,68,47),(10,65,48),(11,66,45),(12,67,46),(13,30,34),(14,31,35),(15,32,36),(16,29,33),(17,72,26),(18,69,27),(19,70,28),(20,71,25),(49,64,54),(50,61,55),(51,62,56),(52,63,53)], [(1,20),(2,17),(3,18),(4,19),(5,11),(6,12),(7,9),(8,10),(13,49),(14,50),(15,51),(16,52),(21,45),(22,46),(23,47),(24,48),(25,40),(26,37),(27,38),(28,39),(29,53),(30,54),(31,55),(32,56),(33,63),(34,64),(35,61),(36,62),(41,67),(42,68),(43,65),(44,66),(57,70),(58,71),(59,72),(60,69)]])

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6R 6S ··· 6Z 12A ··· 12H 12I ··· 12X 12Y ··· 12AF order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 9 9 9 9 1 1 6 6 6 6 1 1 1 1 9 9 9 9 1 ··· 1 6 ··· 6 9 ··· 9 1 ··· 1 6 ··· 6 9 ··· 9

80 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 C4×S3 He3⋊C2 C2×He3⋊C2 C2×He3⋊C2 C4×He3⋊C2 kernel C2×C4×He3⋊C2 C4×He3⋊C2 C2×He3⋊3C4 C2×C4×He3 C22×He3⋊C2 C2×He3⋊C2 C6×C12 C3×C12 C62 C3×C6 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 8 4 8 4 16 4 8 4 16

Matrix representation of C2×C4×He3⋊C2 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 12 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 8
,
 0 12 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 12 12 8 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 1 0 0 0 0 0 10 10 11 0 0 9 0 0 0 0 4 3 3
,
 0 12 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 1 1 5 0 0 0 0 12

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[0,1,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,8,1],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,10,9,4,0,0,10,0,3,0,0,11,0,3],[0,12,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,5,12] >;

C2×C4×He3⋊C2 in GAP, Magma, Sage, TeX

C_2\times C_4\times {\rm He}_3\rtimes C_2
% in TeX

G:=Group("C2xC4xHe3:C2");
// GroupNames label

G:=SmallGroup(432,385);
// by ID

G=gap.SmallGroup(432,385);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,58,1124,4037,537]);
// Polycyclic

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

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