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## G = C2×C32⋊C12order 216 = 23·33

### Direct product of C2 and C32⋊C12

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  C623C12  C62.8D6  C2×C4×C32⋊C6  C62.13D6
C2×C32⋊C12 is a maximal quotient of
He37M4(2)  C62.20D6  C623C12

40 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 6A 6B 6C 6D ··· 6I 6J ··· 6R 12A ··· 12H order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 6 6 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 3 3 6 6 6 9 9 9 9 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 6 6 6 type + + + + - + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 C32⋊C6 C32⋊C12 C2×C32⋊C6 kernel C2×C32⋊C12 C32⋊C12 C22×He3 C2×C3⋊Dic3 C2×He3 C3⋊Dic3 C62 C3×C6 C62 C3×C6 C3×C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 2 1 2 4 2 1 2 1

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

 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 0 0 2 1 0 0 0 0 4 4 0 0 1 2 0 0 0 0 12 0 0 0 3 3 0 0 0 0 12 12 0 0 3 3 0 0 0 0 0 0 12 12 9 9 0 0 0 0 10 10 1 0 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 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 10 0 0 1 0 0 0 0 0 0 0 3 12 12 0 0 0 0 0 0 4 0 0 0 0 1 0 0 0 0 0 9 0 0 12 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 6 6 3 6 3 0 0 0 0 0 3 3 10 3 10 0 0 0 0 4 10 12 3 9 10 0 0 0 0 10 9 9 10 6 0 0 0 0 0 10 1 11 0 1 10 0 0 0 0 4 5 4 3 11 0

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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