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## G = C2×He3⋊3C4order 216 = 23·33

### Direct product of C2 and He3⋊3C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C2×He3⋊3C4
 Chief series C1 — C3 — C32 — He3 — C2×He3 — He3⋊3C4 — C2×He3⋊3C4
 Lower central He3 — C2×He3⋊3C4
 Upper central C1 — C2×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, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, Dic3, C12, C2×C6, C2×C6, C3×C6, C2×Dic3, C2×C12, He3, C3×Dic3, C62, C2×He3, C2×He3, C6×Dic3, He33C4, C22×He3, C2×He33C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C3⋊S3, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, He3⋊C2, C2×C3⋊Dic3, He33C4, 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 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 6A ··· 6F 6G ··· 6R 12A ··· 12H order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 1 1 6 6 6 6 9 9 9 9 1 ··· 1 6 ··· 6 9 ··· 9

40 irreducible representations

 dim 1 1 1 1 2 2 2 3 3 3 type + + + + - + image C1 C2 C2 C4 S3 Dic3 D6 He3⋊C2 He3⋊3C4 C2×He3⋊C2 kernel C2×He3⋊3C4 He3⋊3C4 C22×He3 C2×He3 C62 C3×C6 C3×C6 C22 C2 C2 # reps 1 2 1 4 4 8 4 4 8 4

Matrix representation of C2×He33C4 in GL5(𝔽13)

 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 12 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 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 12 0 0 0 1 12 0 0 0 0 0 0 0 3 0 0 1 0 0 0 0 0 9 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

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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