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G = C2×C32⋊C12  order 216 = 23·33

Direct product of C2 and C32⋊C12

direct product, metabelian, supersoluble, monomial

Aliases: C2×C32⋊C12, C62.2C6, C62.1S3, (C3×C6)⋊C12, He3⋊6(C2×C4), C6.17(S3×C6), (C2×He3)⋊2C4, C3⋊Dic3⋊3C6, (C3×C6)⋊1Dic3, (C3×C6).12D6, C32⋊2(C2×C12), C3.2(C6×Dic3), C6.5(C3×Dic3), C22.(C32⋊C6), C32⋊2(C2×Dic3), (C22×He3).1C2, (C2×He3).9C22, (C2×C3⋊Dic3)⋊C3, (C3×C6).4(C2×C6), (C2×C6).12(C3×S3), C2.2(C2×C32⋊C6), SmallGroup(216,59)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2×C32⋊C12
Chief series
C1 — C3 — C32 — C3×C6 — C2×He3 — C32⋊C12 — C2×C32⋊C12
Lower central
C32 — C2×C32⋊C12
Upper central
C1 — C22

Generators and relations for C2×C32⋊C12
 G = < a,b,c,d | a2=b3=c3=d12=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=c-1 >

Subgroups: 200 in 66 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4 [×2], C22, C6, C6 [×2], C6 [×9], C2×C4, C32 [×2], C32, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×3], C3×C6 [×2], C3×C6 [×4], C3×C6 [×3], C2×Dic3 [×2], C2×C12, He3, C3×Dic3 [×2], C3⋊Dic3 [×2], C62 [×2], C62, C2×He3, C2×He3 [×2], C6×Dic3, C2×C3⋊Dic3, C32⋊C12 [×2], C22×He3, C2×C32⋊C12
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, C3×S3, C2×Dic3, C2×C12, C3×Dic3 [×2], S3×C6, C32⋊C6, C6×Dic3, C32⋊C12 [×2], C2×C32⋊C6, C2×C32⋊C12

Smallest permutation representation of C2×C32⋊C12
►On 72 points
Generators in S72
(1 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(25 56)(26 57)(27 58)(28 59)(29 60)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 61)(47 62)(48 63)

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

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

(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)| (1,21)(2,22)(3,23)(4,24)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,61)(47,62)(48,63), (1,64,72)(2,6,33)(3,11,30)(4,63,67)(5,36,9)(7,70,66)(8,12,27)(10,69,61)(13,55,17)(14,52,22)(15,43,39)(16,20,58)(18,42,46)(19,49,23)(21,37,45)(24,48,40)(25,29,65)(26,34,62)(28,68,32)(31,35,71)(38,56,60)(41,51,59)(44,50,54)(47,57,53), (1,36,68)(2,69,25)(3,26,70)(4,71,27)(5,28,72)(6,61,29)(7,30,62)(8,63,31)(9,32,64)(10,65,33)(11,34,66)(12,67,35)(13,59,45)(14,46,60)(15,49,47)(16,48,50)(17,51,37)(18,38,52)(19,53,39)(20,40,54)(21,55,41)(22,42,56)(23,57,43)(24,44,58), (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( (1,21)(2,22)(3,23)(4,24)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,61)(47,62)(48,63), (1,64,72)(2,6,33)(3,11,30)(4,63,67)(5,36,9)(7,70,66)(8,12,27)(10,69,61)(13,55,17)(14,52,22)(15,43,39)(16,20,58)(18,42,46)(19,49,23)(21,37,45)(24,48,40)(25,29,65)(26,34,62)(28,68,32)(31,35,71)(38,56,60)(41,51,59)(44,50,54)(47,57,53), (1,36,68)(2,69,25)(3,26,70)(4,71,27)(5,28,72)(6,61,29)(7,30,62)(8,63,31)(9,32,64)(10,65,33)(11,34,66)(12,67,35)(13,59,45)(14,46,60)(15,49,47)(16,48,50)(17,51,37)(18,38,52)(19,53,39)(20,40,54)(21,55,41)(22,42,56)(23,57,43)(24,44,58), (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([(1,21),(2,22),(3,23),(4,24),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20),(25,56),(26,57),(27,58),(28,59),(29,60),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,61),(47,62),(48,63)], [(1,64,72),(2,6,33),(3,11,30),(4,63,67),(5,36,9),(7,70,66),(8,12,27),(10,69,61),(13,55,17),(14,52,22),(15,43,39),(16,20,58),(18,42,46),(19,49,23),(21,37,45),(24,48,40),(25,29,65),(26,34,62),(28,68,32),(31,35,71),(38,56,60),(41,51,59),(44,50,54),(47,57,53)], [(1,36,68),(2,69,25),(3,26,70),(4,71,27),(5,28,72),(6,61,29),(7,30,62),(8,63,31),(9,32,64),(10,65,33),(11,34,66),(12,67,35),(13,59,45),(14,46,60),(15,49,47),(16,48,50),(17,51,37),(18,38,52),(19,53,39),(20,40,54),(21,55,41),(22,42,56),(23,57,43),(24,44,58)], [(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×C32⋊C12 is a maximal subgroup of
 He3⋊C42  C62.D6  C62.3D6  C62.4D6  C62.5D6  C62.19D6  C62.20D6  C62.21D6  C62⋊3C12  C62.8D6  C2×C4×C32⋊C6  C62.13D6
C2×C32⋊C12 is a maximal quotient of
 He3⋊7M4(2)  C62.20D6  C62⋊3C12

40 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F4A4B4C4D6A6B6C6D···6I6J···6R12A···12H
order122233333344446666···66···612···12
size111123366699992223···36···69···9

40 irreducible representations

dim11111111222222666
type++++-++-+
imageC1C2C2C3C4C6C6C12S3Dic3D6C3×S3C3×Dic3S3×C6C32⋊C6C32⋊C12C2×C32⋊C6
kernelC2×C32⋊C12C32⋊C12C22×He3C2×C3⋊Dic3C2×He3C3⋊Dic3C62C3×C6C62C3×C6C3×C6C2×C6C6C6C22C2C2
# reps12124428121242121

Matrix representation of C2×C32⋊C12 ►in GL10(𝔽13)

12000000000
01200000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
9000000000
0300000000
0090000000
0003000000
0000440021
0000440012
00001200033
000012120033
000000121299
000010101099
,
1000000000
0100000000
0010000000
0001000000
00001210000
00001200000
00001000100
000003121200
0000400001
000009001212
,
0800000000
8000000000
0008000000
0080000000
0000066363
000003310310
0000410123910
000010991060
0000101110110
00004543110

G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,4,4,12,12,0,10,0,0,0,0,4,4,0,12,0,10,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,2,1,3,3,9,9,0,0,0,0,1,2,3,3,9,9],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,10,0,4,0,0,0,0,0,1,0,0,3,0,9,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,4,10,10,4,0,0,0,0,6,3,10,9,1,5,0,0,0,0,6,3,12,9,11,4,0,0,0,0,3,10,3,10,0,3,0,0,0,0,6,3,9,6,1,11,0,0,0,0,3,10,10,0,10,0] >;
 

C2×C32⋊C12 in GAP, Magma, Sage, TeX 

C_2\times C_3^2\rtimes C_{12}
 
% in TeX
 
G:=Group("C2xC3^2:C12");
 
// GroupNames label
 
G:=SmallGroup(216,59);
 
// by ID
 
G=gap.SmallGroup(216,59);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,1444,736,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^3=c^3=d^12=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=c^-1>;
 
// generators/relations
 

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