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## G = C2×He3⋊4D4order 432 = 24·33

### Direct product of C2 and He3⋊4D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×He3⋊4D4
 Chief series C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — C22×C32⋊C6 — C2×He3⋊4D4
 Lower central C32 — C3×C6 — C2×He3⋊4D4
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×He34D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 1097 in 205 conjugacy classes, 62 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C12, C12, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, He3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C2×D12, C6×D4, C32⋊C6, C2×He3, C2×He3, C3×D12, C12⋊S3, C6×C12, C6×C12, S3×C2×C6, C22×C3⋊S3, C4×He3, C2×C32⋊C6, C2×C32⋊C6, C22×He3, C6×D12, C2×C12⋊S3, He34D4, C2×C4×He3, C22×C32⋊C6, C2×He34D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, S3×C6, C2×D12, C6×D4, C32⋊C6, C3×D12, S3×C2×C6, C2×C32⋊C6, C6×D12, He34D4, C22×C32⋊C6, C2×He34D4

Smallest permutation representation of C2×He34D4
On 72 points
Generators in S72
(1 41)(2 42)(3 43)(4 44)(5 48)(6 45)(7 46)(8 47)(9 63)(10 64)(11 61)(12 62)(13 56)(14 53)(15 54)(16 55)(17 37)(18 38)(19 39)(20 40)(21 59)(22 60)(23 57)(24 58)(25 65)(26 66)(27 67)(28 68)(29 52)(30 49)(31 50)(32 51)(33 69)(34 70)(35 71)(36 72)
(1 29 14)(2 30 15)(3 31 16)(4 32 13)(17 67 22)(18 68 23)(19 65 24)(20 66 21)(25 58 39)(26 59 40)(27 60 37)(28 57 38)(41 52 53)(42 49 54)(43 50 55)(44 51 56)
(1 29 14)(2 30 15)(3 31 16)(4 32 13)(5 10 33)(6 11 34)(7 12 35)(8 9 36)(17 22 67)(18 23 68)(19 24 65)(20 21 66)(25 39 58)(26 40 59)(27 37 60)(28 38 57)(41 52 53)(42 49 54)(43 50 55)(44 51 56)(45 61 70)(46 62 71)(47 63 72)(48 64 69)
(1 35 26)(2 36 27)(3 33 28)(4 34 25)(5 38 31)(6 39 32)(7 40 29)(8 37 30)(9 60 15)(10 57 16)(11 58 13)(12 59 14)(17 49 47)(18 50 48)(19 51 45)(20 52 46)(21 53 62)(22 54 63)(23 55 64)(24 56 61)(41 71 66)(42 72 67)(43 69 68)(44 70 65)
(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 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 29)(14 32)(15 31)(16 30)(17 23)(18 22)(19 21)(20 24)(25 26)(27 28)(33 36)(34 35)(37 57)(38 60)(39 59)(40 58)(41 44)(42 43)(45 62)(46 61)(47 64)(48 63)(49 55)(50 54)(51 53)(52 56)(65 66)(67 68)(69 72)(70 71)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D ··· 6I 6J ··· 6R 6S ··· 6Z 12A 12B 12C 12D 12E ··· 12T order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 6 6 6 6 ··· 6 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 1 1 18 18 18 18 2 3 3 6 6 6 2 2 2 2 2 3 ··· 3 6 ··· 6 18 ··· 18 2 2 2 2 6 ··· 6

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D6 C3×S3 D12 C3×D4 S3×C6 S3×C6 C3×D12 C32⋊C6 C2×C32⋊C6 C2×C32⋊C6 He3⋊4D4 kernel C2×He3⋊4D4 He3⋊4D4 C2×C4×He3 C22×C32⋊C6 C2×C12⋊S3 C12⋊S3 C6×C12 C22×C3⋊S3 C6×C12 C2×He3 C3×C12 C62 C2×C12 C3×C6 C3×C6 C12 C2×C6 C6 C2×C4 C4 C22 C2 # reps 1 4 1 2 2 8 2 4 1 2 2 1 2 4 4 4 2 8 1 2 1 4

Matrix representation of C2×He34D4 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 10 6 0 0 0 0 0 0 7 3 0 0 0 0 0 0 0 0 10 6 0 0 0 0 0 0 7 3 0 0 0 0 0 0 0 0 10 6 0 0 0 0 0 0 7 3 0 0 0 0 0 0 0 0 10 6 0 0 0 0 0 0 7 3
,
 7 3 0 0 0 0 0 0 10 6 0 0 0 0 0 0 0 0 6 10 0 0 0 0 0 0 3 7 0 0 0 0 0 0 0 0 6 10 0 0 0 0 0 0 3 7 0 0 0 0 0 0 0 0 6 10 0 0 0 0 0 0 3 7

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[10,7,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,10,7,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,10,7,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,10,7,0,0,0,0,0,0,6,3],[7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,6,3,0,0,0,0,0,0,10,7,0,0,0,0,0,0,0,0,6,3,0,0,0,0,0,0,10,7,0,0,0,0,0,0,0,0,6,3,0,0,0,0,0,0,10,7] >;

C2×He34D4 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_4D_4
% in TeX

G:=Group("C2xHe3:4D4");
// GroupNames label

G:=SmallGroup(432,350);
// by ID

G=gap.SmallGroup(432,350);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,142,4037,1034,14118]);
// Polycyclic

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

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