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G = C2×He3⋊6D4  order 432 = 24·33

Direct product of C2 and He3⋊6D4

direct product, metabelian, supersoluble, monomial

Aliases: C2×He3⋊6D4, C62⋊12D6, (C2×He3)⋊6D4, (C2×C62)⋊2C6, C62⋊4(C2×C6), He3⋊12(C2×D4), (C2×C62)⋊2S3, C32⋊4(C6×D4), C32⋊7D4⋊4C6, (C23×He3)⋊2C2, C32⋊C12⋊9C22, C23⋊3(C32⋊C6), (C2×He3).28C23, (C22×He3)⋊7C22, (C3×C6)⋊3(C3×D4), C6.46(S3×C2×C6), (C2×C32⋊7D4)⋊C3, C3.2(C6×C3⋊D4), (C3×C6)⋊4(C3⋊D4), (C22×C3⋊S3)⋊3C6, (C2×C6).64(S3×C6), (C2×C3⋊Dic3)⋊4C6, C3⋊Dic3⋊2(C2×C6), C6.34(C3×C3⋊D4), C32⋊5(C2×C3⋊D4), C22⋊4(C2×C32⋊C6), (C2×C32⋊C12)⋊10C2, (C22×C32⋊C6)⋊6C2, (C2×C32⋊C6)⋊9C22, (C22×C6).30(C3×S3), (C3×C6).10(C22×C6), (C3×C6).36(C22×S3), C2.10(C22×C32⋊C6), (C2×C3⋊S3)⋊3(C2×C6), SmallGroup(432,377)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C2×He3⋊6D4
Chief series
C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — C22×C32⋊C6 — C2×He3⋊6D4
Lower central
C32 — C3×C6 — C2×He3⋊6D4
Upper central
C1 — C22 — C23

Generators and relations for C2×He3⋊6D4
 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, ece-1=fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 969 in 221 conjugacy classes, 62 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C2×C3⋊D4, C6×D4, C32⋊C6, C2×He3, C2×He3, C2×He3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, S3×C2×C6, C22×C3⋊S3, C2×C62, C2×C62, C32⋊C12, C2×C32⋊C6, C2×C32⋊C6, C22×He3, C22×He3, C22×He3, C6×C3⋊D4, C2×C32⋊7D4, C2×C32⋊C12, He3⋊6D4, C22×C32⋊C6, C23×He3, C2×He3⋊6D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, S3×C6, C2×C3⋊D4, C6×D4, C32⋊C6, C3×C3⋊D4, S3×C2×C6, C2×C32⋊C6, C6×C3⋊D4, He3⋊6D4, C22×C32⋊C6, C2×He3⋊6D4

Smallest permutation representation of C2×He3⋊6D4
►On 72 points
Generators in S72
(1 40)(2 37)(3 38)(4 39)(5 46)(6 47)(7 48)(8 45)(9 18)(10 19)(11 20)(12 17)(13 35)(14 36)(15 33)(16 34)(21 25)(22 26)(23 27)(24 28)(29 58)(30 59)(31 60)(32 57)(41 62)(42 63)(43 64)(44 61)(49 71)(50 72)(51 69)(52 70)(53 66)(54 67)(55 68)(56 65)

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E3F4A4B6A···6G6H···6M6N···6AL6AM6AN6AO6AP12A12B12C12D
order12222222333333446···66···66···6666612121212
size111122181823366618182···23···36···61818181818181818

62 irreducible representations

dim111111111122222222666
type++++++++++
imageC1C2C2C2C2C3C6C6C6C6S3D4D6C3×S3C3⋊D4C3×D4S3×C6C3×C3⋊D4C32⋊C6C2×C32⋊C6He3⋊6D4
kernelC2×He3⋊6D4C2×C32⋊C12He3⋊6D4C22×C32⋊C6C23×He3C2×C32⋊7D4C2×C3⋊Dic3C32⋊7D4C22×C3⋊S3C2×C62C2×C62C2×He3C62C22×C6C3×C6C3×C6C2×C6C6C23C22C2
# reps114112282212324468134

Matrix representation of C2×He3⋊6D4 ►in GL10(𝔽13)

1000000000
0100000000
00120000000
00012000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
0030000000
0089000000
000000001212
0000000010
000012120000
0000100000
000000121200
0000001000
,
1000000000
0100000000
0010000000
0001000000
0000010000
000012120000
0000000100
000000121200
0000000001
000000001212
,
9000000000
0900000000
0010000000
0001000000
0000001000
0000000100
0000000001
000000001212
000012120000
0000100000
,
8200000000
0500000000
0043000000
0089000000
00001010031010
0000033003
00000310101010
0000300303
00001010101003
0000030330
,
51100000000
12800000000
00910000000
0054000000
00001010031010
0000033003
00000310101010
0000300303
00001010101003
0000030330

G:=sub<GL(10,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,8,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0],[8,0,0,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,0,0,4,8,0,0,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,0,0,10,0,0,3,10,0,0,0,0,0,10,3,3,0,10,3,0,0,0,0,0,3,10,0,10,0,0,0,0,0,3,0,10,3,10,3,0,0,0,0,10,0,10,0,0,3,0,0,0,0,10,3,10,3,3,0],[5,12,0,0,0,0,0,0,0,0,11,8,0,0,0,0,0,0,0,0,0,0,9,5,0,0,0,0,0,0,0,0,10,4,0,0,0,0,0,0,0,0,0,0,10,0,0,3,10,0,0,0,0,0,10,3,3,0,10,3,0,0,0,0,0,3,10,0,10,0,0,0,0,0,3,0,10,3,10,3,0,0,0,0,10,0,10,0,0,3,0,0,0,0,10,3,10,3,3,0] >;
 

C2×He3⋊6D4 in GAP, Magma, Sage, TeX 

C_2\times {\rm He}_3\rtimes_6D_4
 
% in TeX
 
G:=Group("C2xHe3:6D4");
 
// GroupNames label
 
G:=SmallGroup(432,377);
 
// by ID
 
G=gap.SmallGroup(432,377);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,4037,1034,14118]);
 
// 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,e*b*e^-1=f*b*f=b^-1,c*d=d*c,e*c*e^-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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