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G = C2×He3⋊2D4  order 432 = 24·33

Direct product of C2 and He3⋊2D4

direct product, non-abelian, supersoluble, monomial

Aliases: C2×He3⋊2D4, C62.10D6, He3⋊4(C2×D4), (C2×He3)⋊2D4, C6.6(D6⋊S3), He3⋊3C4⋊3C22, (C2×He3).14C23, C22.10(C32⋊D6), (C22×He3).10C22, C6.88(C2×S32), (C2×C6).55S32, (C2×C3⋊S3)⋊4D6, (C3×C6)⋊1(C3⋊D4), (C22×C3⋊S3)⋊1S3, C32⋊1(C2×C3⋊D4), (C2×He3⋊3C4)⋊4C2, C3.1(C2×D6⋊S3), C2.15(C2×C32⋊D6), (C22×C32⋊C6)⋊1C2, (C2×C32⋊C6)⋊4C22, (C3×C6).14(C22×S3), SmallGroup(432,320)

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×He3⋊2D4
 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, He3⋊3C4, C2×C32⋊C6, C2×C32⋊C6, C22×He3, C2×C3⋊D12, He3⋊2D4, C2×He3⋊3C4, C22×C32⋊C6, C2×He3⋊2D4
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, He3⋊2D4, C2×C32⋊D6, C2×He3⋊2D4

Smallest permutation representation of C2×He3⋊2D4
►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 2A2B2C2D2E2F2G3A3B3C3D4A4B6A6B6C6D···6I6J6K6L6M···6T12A12B12C12D
order122222223333446666···66666···612121212
size1111181818182661218182226···612121218···1818181818

38 irreducible representations

dim111122222444666
type+++++++++-++++
imageC1C2C2C2S3D4D6D6C3⋊D4S32D6⋊S3C2×S32C32⋊D6He3⋊2D4C2×C32⋊D6
kernelC2×He3⋊2D4He3⋊2D4C2×He3⋊3C4C22×C32⋊C6C22×C3⋊S3C2×He3C2×C3⋊S3C62C3×C6C2×C6C6C6C22C2C2
# reps141222428121242

Matrix representation of C2×He3⋊2D4 ►in GL10(𝔽13)

1000000000
0100000000
0010000000
0001000000
00001200000
00000120000
00000012000
00000001200
00000000120
00000000012
,
0010000000
0001000000
120120000000
012012000000
0000001000
0000000100
0000000010
0000000001
0000100000
0000010000
,
1000000000
0100000000
0010000000
0001000000
00001210000
00001200000
00000012100
00000012000
00000000121
00000000120
,
9000000000
3300000000
0090000000
0033000000
0000100000
0000010000
00000001200
00000011200
00000000121
00000000120
,
11100000000
11200000000
122122000000
121121000000
0000370000
00006100000
0000000037
00000000610
0000003700
00000061000
,
1000000000
11200000000
120120000000
121121000000
00000120000
00001200000
00000000012
00000000120
00000001200
00000012000

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×He3⋊2D4 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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