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## G = C2×He3⋊2D4order 432 = 24·33

### Direct product of C2 and He3⋊2D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C2×He3⋊2D4
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — He3⋊2D4 — C2×He3⋊2D4
 Lower central He3 — C2×He3 — C2×He3⋊2D4
 Upper central C1 — C22

Generators and relations for C2×He32D4
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, ebe-1=fbf=b-1, cd=dc, ce=ec, fcf=c-1, ede-1=d-1, df=fd, fef=e-1 >

Subgroups: 1299 in 205 conjugacy classes, 45 normal (11 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, He3, C3×Dic3, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C2×D12, C2×C3⋊D4, C32⋊C6, C2×He3, C2×He3, C3⋊D12, C6×Dic3, S3×C2×C6, C22×C3⋊S3, He33C4, C2×C32⋊C6, C2×C32⋊C6, C22×He3, C2×C3⋊D12, He32D4, C2×He33C4, C22×C32⋊C6, C2×He32D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, S32, C2×C3⋊D4, D6⋊S3, C2×S32, C32⋊D6, C2×D6⋊S3, He32D4, C2×C32⋊D6, C2×He32D4

Smallest permutation representation of C2×He32D4
On 72 points
Generators in S72
(1 47)(2 48)(3 45)(4 46)(5 70)(6 71)(7 72)(8 69)(9 35)(10 36)(11 33)(12 34)(13 24)(14 21)(15 22)(16 23)(17 32)(18 29)(19 30)(20 31)(25 49)(26 50)(27 51)(28 52)(37 41)(38 42)(39 43)(40 44)(53 65)(54 66)(55 67)(56 68)(57 63)(58 64)(59 61)(60 62)
(1 26 64)(2 61 27)(3 28 62)(4 63 25)(5 32 41)(6 42 29)(7 30 43)(8 44 31)(9 53 24)(10 21 54)(11 55 22)(12 23 56)(13 35 65)(14 66 36)(15 33 67)(16 68 34)(17 37 70)(18 71 38)(19 39 72)(20 69 40)(45 52 60)(46 57 49)(47 50 58)(48 59 51)
(1 15 29)(2 16 30)(3 13 31)(4 14 32)(5 25 36)(6 26 33)(7 27 34)(8 28 35)(9 69 52)(10 70 49)(11 71 50)(12 72 51)(17 46 21)(18 47 22)(19 48 23)(20 45 24)(37 57 54)(38 58 55)(39 59 56)(40 60 53)(41 63 66)(42 64 67)(43 61 68)(44 62 65)
(5 36 25)(6 26 33)(7 34 27)(8 28 35)(9 69 52)(10 49 70)(11 71 50)(12 51 72)(37 57 54)(38 55 58)(39 59 56)(40 53 60)(41 63 66)(42 67 64)(43 61 68)(44 65 62)
(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 47)(2 46)(3 45)(4 48)(5 56)(6 55)(7 54)(8 53)(9 44)(10 43)(11 42)(12 41)(13 20)(14 19)(15 18)(16 17)(21 30)(22 29)(23 32)(24 31)(25 59)(26 58)(27 57)(28 60)(33 38)(34 37)(35 40)(36 39)(49 61)(50 64)(51 63)(52 62)(65 69)(66 72)(67 71)(68 70)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D ··· 6I 6J 6K 6L 6M ··· 6T 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 6 6 6 6 ··· 6 6 6 6 6 ··· 6 12 12 12 12 size 1 1 1 1 18 18 18 18 2 6 6 12 18 18 2 2 2 6 ··· 6 12 12 12 18 ··· 18 18 18 18 18

38 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 4 4 6 6 6 type + + + + + + + + + - + + + + image C1 C2 C2 C2 S3 D4 D6 D6 C3⋊D4 S32 D6⋊S3 C2×S32 C32⋊D6 He3⋊2D4 C2×C32⋊D6 kernel C2×He3⋊2D4 He3⋊2D4 C2×He3⋊3C4 C22×C32⋊C6 C22×C3⋊S3 C2×He3 C2×C3⋊S3 C62 C3×C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 1 2 2 2 4 2 8 1 2 1 2 4 2

Matrix representation of C2×He32D4 in GL10(𝔽13)

 1 0 0 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 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12
,
 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 12 0 0 0 0 0 0 0 0 12 0 12 0 0 0 0 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 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 1 0 0 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 0 0 1 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0
,
 9 0 0 0 0 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 3 3 0 0 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 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0
,
 1 11 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 12 2 12 2 0 0 0 0 0 0 12 1 12 1 0 0 0 0 0 0 0 0 0 0 3 7 0 0 0 0 0 0 0 0 6 10 0 0 0 0 0 0 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
,
 1 0 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 12 0 12 0 0 0 0 0 0 0 12 1 12 1 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0

G:=sub<GL(10,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0],[9,3,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,9,3,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0],[1,1,12,12,0,0,0,0,0,0,11,12,2,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,7,10,0,0],[1,1,12,12,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0] >;

C2×He32D4 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_2D_4
% in TeX

G:=Group("C2xHe3:2D4");
// GroupNames label

G:=SmallGroup(432,320);
// by ID

G=gap.SmallGroup(432,320);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,571,4037,537,14118,7069]);
// 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,e*b*e^-1=f*b*f=b^-1,c*d=d*c,c*e=e*c,f*c*f=c^-1,e*d*e^-1=d^-1,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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