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G = C2×He33C4order 216 = 23·33

Direct product of C2 and He33C4

direct product, non-abelian, supersoluble, monomial

Aliases: C2×He33C4, C62.4S3, He37(C2×C4), (C2×He3)⋊3C4, (C3×C6)⋊2Dic3, (C3×C6).19D6, C6.5(C3⋊Dic3), C323(C2×Dic3), C22.(He3⋊C2), (C22×He3).2C2, (C2×He3).14C22, C6.20(C2×C3⋊S3), (C2×C6).9(C3⋊S3), C3.2(C2×C3⋊Dic3), C2.2(C2×He3⋊C2), SmallGroup(216,71)

Series: Derived Chief Lower central Upper central

C1C3He3 — C2×He33C4
C1C3C32He3C2×He3He33C4 — C2×He33C4
He3 — C2×He33C4
C1C2×C6

Generators and relations for C2×He33C4
 G = < a,b,c,d,e | a2=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 224 in 88 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4 [×2], C22, C6, C6 [×2], C6 [×12], C2×C4, C32 [×4], Dic3 [×8], C12 [×2], C2×C6, C2×C6 [×4], C3×C6 [×12], C2×Dic3 [×4], C2×C12, He3, C3×Dic3 [×8], C62 [×4], C2×He3, C2×He3 [×2], C6×Dic3 [×4], He33C4 [×2], C22×He3, C2×He33C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, Dic3 [×8], D6 [×4], C3⋊S3, C2×Dic3 [×4], C3⋊Dic3 [×2], C2×C3⋊S3, He3⋊C2, C2×C3⋊Dic3, He33C4 [×2], C2×He3⋊C2, C2×He33C4

Smallest permutation representation of C2×He33C4
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 62 29)(2 30 63)(3 64 31)(4 32 61)(5 27 52)(6 49 28)(7 25 50)(8 51 26)(9 33 56)(10 53 34)(11 35 54)(12 55 36)(13 67 20)(14 17 68)(15 65 18)(16 19 66)(21 72 48)(22 45 69)(23 70 46)(24 47 71)(37 59 42)(38 43 60)(39 57 44)(40 41 58)
(1 26 15)(2 27 16)(3 28 13)(4 25 14)(5 66 63)(6 67 64)(7 68 61)(8 65 62)(9 58 69)(10 59 70)(11 60 71)(12 57 72)(17 32 50)(18 29 51)(19 30 52)(20 31 49)(21 36 39)(22 33 40)(23 34 37)(24 35 38)(41 45 56)(42 46 53)(43 47 54)(44 48 55)
(1 29 65)(2 66 30)(3 31 67)(4 68 32)(5 19 16)(6 13 20)(7 17 14)(8 15 18)(9 45 33)(10 34 46)(11 47 35)(12 36 48)(21 44 72)(22 69 41)(23 42 70)(24 71 43)(25 61 50)(26 51 62)(27 63 52)(28 49 64)(37 53 59)(38 60 54)(39 55 57)(40 58 56)
(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)

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

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

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

C2×He33C4 is a maximal subgroup of
He3⋊C42  C62.D6  C62.3D6  C62.4D6  C62.29D6  C62.30D6  C62.31D6  C624Dic3  He34M4(2)  C62.9D6  C2×C4×He3⋊C2  C62.16D6
C2×He33C4 is a maximal quotient of
He38M4(2)  C62.30D6  C624Dic3

40 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F4A4B4C4D6A···6F6G···6R12A···12H
order122233333344446···66···612···12
size111111666699991···16···69···9

40 irreducible representations

dim1111222333
type++++-+
imageC1C2C2C4S3Dic3D6He3⋊C2He33C4C2×He3⋊C2
kernelC2×He33C4He33C4C22×He3C2×He3C62C3×C6C3×C6C22C2C2
# reps1214484484

Matrix representation of C2×He33C4 in GL5(𝔽13)

10000
01000
001200
000120
000012
,
012000
112000
00010
00001
00100
,
10000
01000
00300
00030
00003
,
012000
112000
00003
00100
00090
,
05000
50000
00010
00100
00001

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

C2×He33C4 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_3C_4
% in TeX

G:=Group("C2xHe3:3C4");
// GroupNames label

G:=SmallGroup(216,71);
// by ID

G=gap.SmallGroup(216,71);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,387,1444,382]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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