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G = C2×He3⋊4D4  order 432 = 24·33

Direct product of C2 and He3⋊4D4

direct product, metabelian, supersoluble, monomial

Aliases: C2×He3⋊4D4, C62.35D6, (C6×C12)⋊5S3, (C6×C12)⋊3C6, (C3×C12)⋊6D6, (C3×C6)⋊3D12, He3⋊8(C2×D4), (C2×He3)⋊4D4, C3.2(C6×D12), C12⋊S3⋊5C6, C32⋊2(C6×D4), C12.77(S3×C6), C6.17(C3×D12), C32⋊4(C2×D12), C62.11(C2×C6), (C4×He3)⋊6C22, (C2×He3).21C23, (C22×He3).28C22, (C3×C6)⋊2(C3×D4), C6.25(S3×C2×C6), (C2×C4×He3)⋊6C2, (C2×C12⋊S3)⋊C3, (C3×C12)⋊2(C2×C6), C4⋊2(C2×C32⋊C6), (C22×C3⋊S3)⋊1C6, (C2×C6).55(S3×C6), (C2×C4)⋊2(C32⋊C6), (C2×C12).17(C3×S3), (C3×C6).3(C22×C6), (C2×C32⋊C6)⋊7C22, (C22×C32⋊C6)⋊4C2, (C3×C6).21(C22×S3), C2.4(C22×C32⋊C6), C22.10(C2×C32⋊C6), (C2×C3⋊S3)⋊1(C2×C6), SmallGroup(432,350)

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×He3⋊4D4
 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, He3⋊4D4, C2×C4×He3, C22×C32⋊C6, C2×He3⋊4D4
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, He3⋊4D4, C22×C32⋊C6, C2×He3⋊4D4

Smallest permutation representation of C2×He3⋊4D4
►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 2A2B2C2D2E2F2G3A3B3C3D3E3F4A4B6A6B6C6D···6I6J···6R6S···6Z12A12B12C12D12E···12T
order12222222333333446666···66···66···61212121212···12
size111118181818233666222223···36···618···1822226···6

62 irreducible representations

dim1111111122222222226666
type+++++++++++++
imageC1C2C2C2C3C6C6C6S3D4D6D6C3×S3D12C3×D4S3×C6S3×C6C3×D12C32⋊C6C2×C32⋊C6C2×C32⋊C6He3⋊4D4
kernelC2×He3⋊4D4He3⋊4D4C2×C4×He3C22×C32⋊C6C2×C12⋊S3C12⋊S3C6×C12C22×C3⋊S3C6×C12C2×He3C3×C12C62C2×C12C3×C6C3×C6C12C2×C6C6C2×C4C4C22C2
# reps1412282412212444281214

Matrix representation of C2×He3⋊4D4 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
00001000
00000100
00000010
00000001
,
012000000
112000000
000120000
001120000
00001000
00000100
000000121
000000120
,
10000000
01000000
000120000
001120000
000001200
000011200
000000012
000000112
,
10000000
01000000
00000010
00000001
00100000
00010000
00001000
00000100
,
106000000
73000000
001060000
00730000
000010600
00007300
000000106
00000073
,
73000000
106000000
006100000
00370000
000061000
00003700
000000610
00000037

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×He3⋊4D4 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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