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G = C2×He3⋊3D4  order 432 = 24·33

Direct product of C2 and He3⋊3D4

direct product, non-abelian, supersoluble, monomial

Aliases: C2×He3⋊3D4, C62.12D6, (C3×C6)⋊D12, He3⋊5(C2×D4), (C2×He3)⋊3D4, C3⋊Dic3⋊7D6, C32⋊2(C2×D12), C32⋊C12⋊4C22, C6.36(C3⋊D12), (C2×He3).16C23, C22.12(C32⋊D6), (C22×He3).12C22, C6.90(C2×S32), (C2×C6).57S32, (C2×C3⋊S3)⋊5D6, (C3×C6)⋊2(C3⋊D4), (C22×C3⋊S3)⋊2S3, (C2×C3⋊Dic3)⋊6S3, C32⋊2(C2×C3⋊D4), (C2×C32⋊C12)⋊5C2, C3.3(C2×C3⋊D12), C2.16(C2×C32⋊D6), (C22×C32⋊C6)⋊2C2, (C2×C32⋊C6)⋊5C22, (C3×C6).16(C22×S3), (C22×He3⋊C2)⋊1C2, (C2×He3⋊C2)⋊3C22, SmallGroup(432,322)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2×He3 — C2×He3⋊3D4
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — He3⋊3D4 — C2×He3⋊3D4
Lower central
He3 — C2×He3 — C2×He3⋊3D4
Upper central
C1 — C22

Generators and relations for C2×He3⋊3D4
 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, ece-1=c-1, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 1363 in 221 conjugacy classes, 45 normal (21 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, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C2×D12, C2×C3⋊D4, C32⋊C6, He3⋊C2, C2×He3, C2×He3, D6⋊S3, C3⋊D12, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C22×C3⋊S3, C32⋊C12, C2×C32⋊C6, C2×C32⋊C6, C2×He3⋊C2, C2×He3⋊C2, C22×He3, C2×D6⋊S3, C2×C3⋊D12, He3⋊3D4, C2×C32⋊C12, C22×C32⋊C6, C22×He3⋊C2, C2×He3⋊3D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, S32, C2×D12, C2×C3⋊D4, C3⋊D12, C2×S32, C32⋊D6, C2×C3⋊D12, He3⋊3D4, C2×C32⋊D6, C2×He3⋊3D4

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A4B6A6B6C6D···6I6J6K6L6M···6T12A12B12C12D
order122222223333446666···66666···612121212
size1111181818182661218182226···612121218···1818181818

38 irreducible representations

dim1111122222222444666
type+++++++++++++++++
imageC1C2C2C2C2S3S3D4D6D6D6D12C3⋊D4S32C3⋊D12C2×S32C32⋊D6He3⋊3D4C2×C32⋊D6
kernelC2×He3⋊3D4He3⋊3D4C2×C32⋊C12C22×C32⋊C6C22×He3⋊C2C2×C3⋊Dic3C22×C3⋊S3C2×He3C3⋊Dic3C2×C3⋊S3C62C3×C6C3×C6C2×C6C6C6C22C2C2
# reps1411111222244121242

Matrix representation of C2×He3⋊3D4 ►in GL14(𝔽13)

120000000000000
012000000000000
001200000000000
000120000000000
00001000000000
00000100000000
00000010000000
00000001000000
00000000100000
00000000010000
00000000001000
00000000000100
00000000000010
00000000000001
,
1201200000000000
0120120000000000
10000000000000
01000000000000
0000120120000000
0000012012000000
00001000000000
00000100000000
00000000000010
00000000000001
00000000100000
00000000010000
00000000001000
00000000000100
,
10000000000000
01000000000000
00100000000000
00010000000000
00001000000000
00000100000000
00000010000000
00000001000000
000000000120000
000000001120000
000000000001200
000000000011200
000000000000012
000000000000112
,
012000000000000
112000000000000
000120000000000
001120000000000
000001200000000
000011200000000
000000012000000
000000112000000
00000000100000
00000000010000
000000000001200
000000000011200
000000000000121
000000000000120
,
010070000000000
100700000000000
06030000000000
60300000000000
000001007000000
000010070000000
00000603000000
00006030000000
000000000120000
000000001200000
000000000001200
000000000012000
000000000000012
000000000000120
,
012000000000000
120000000000000
01010000000000
10100000000000
000001200000000
000012000000000
00000101000000
00001010000000
00000000100000
00000000010000
00000000000010
00000000000001
00000000001000
00000000000100

G:=sub<GL(14,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,12],[0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0],[0,10,0,6,0,0,0,0,0,0,0,0,0,0,10,0,6,0,0,0,0,0,0,0,0,0,0,0,0,7,0,3,0,0,0,0,0,0,0,0,0,0,7,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,6,0,0,0,0,0,0,0,0,0,0,10,0,6,0,0,0,0,0,0,0,0,0,0,0,0,7,0,3,0,0,0,0,0,0,0,0,0,0,7,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0],[0,12,0,1,0,0,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
 

C2×He3⋊3D4 in GAP, Magma, Sage, TeX 

C_2\times {\rm He}_3\rtimes_3D_4
 
% in TeX
 
G:=Group("C2xHe3:3D4");
 
// GroupNames label
 
G:=SmallGroup(432,322);
 
// by ID
 
G=gap.SmallGroup(432,322);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,571,4037,537,14118,7069]);
 
// 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,b*e=e*b,f*b*f=b^-1,c*d=d*c,e*c*e^-1=c^-1,c*f=f*c,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

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