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## G = C6×C32⋊7D4order 432 = 24·33

### Direct product of C6 and C32⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C6×C32⋊7D4
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C6×C3⋊S3 — C2×C6×C3⋊S3 — C6×C32⋊7D4
 Lower central C32 — C3×C6 — C6×C32⋊7D4
 Upper central C1 — C2×C6 — C22×C6

Generators and relations for C6×C327D4
G = < a,b,c,d,e | a6=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1284 in 452 conjugacy classes, 118 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C22×C6, C33, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C2×C3⋊D4, C6×D4, C3×C3⋊S3, C32×C6, C32×C6, C32×C6, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C327D4, S3×C2×C6, C22×C3⋊S3, C2×C62, C2×C62, C2×C62, C3×C3⋊Dic3, C6×C3⋊S3, C6×C3⋊S3, C3×C62, C3×C62, C3×C62, C6×C3⋊D4, C2×C327D4, C6×C3⋊Dic3, C3×C327D4, C2×C6×C3⋊S3, C63, C6×C327D4
Quotients:

Smallest permutation representation of C6×C327D4
On 72 points
Generators in S72
(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 27 16)(2 28 17)(3 29 18)(4 30 13)(5 25 14)(6 26 15)(7 34 39)(8 35 40)(9 36 41)(10 31 42)(11 32 37)(12 33 38)(19 70 59)(20 71 60)(21 72 55)(22 67 56)(23 68 57)(24 69 58)(43 51 62)(44 52 63)(45 53 64)(46 54 65)(47 49 66)(48 50 61)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)
(1 71 35 44)(2 72 36 45)(3 67 31 46)(4 68 32 47)(5 69 33 48)(6 70 34 43)(7 51 15 59)(8 52 16 60)(9 53 17 55)(10 54 18 56)(11 49 13 57)(12 50 14 58)(19 39 62 26)(20 40 63 27)(21 41 64 28)(22 42 65 29)(23 37 66 30)(24 38 61 25)
(1 47)(2 48)(3 43)(4 44)(5 45)(6 46)(7 56)(8 57)(9 58)(10 59)(11 60)(12 55)(13 52)(14 53)(15 54)(16 49)(17 50)(18 51)(19 42)(20 37)(21 38)(22 39)(23 40)(24 41)(25 64)(26 65)(27 66)(28 61)(29 62)(30 63)(31 70)(32 71)(33 72)(34 67)(35 68)(36 69)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3N 4A 4B 6A ··· 6F 6G ··· 6CP 6CQ 6CR 6CS 6CT 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 size 1 1 1 1 2 2 18 18 1 1 2 ··· 2 18 18 1 ··· 1 2 ··· 2 18 18 18 18 18 18 18 18

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D6 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×C3⋊D4 kernel C6×C32⋊7D4 C6×C3⋊Dic3 C3×C32⋊7D4 C2×C6×C3⋊S3 C63 C2×C32⋊7D4 C2×C3⋊Dic3 C32⋊7D4 C22×C3⋊S3 C2×C62 C2×C62 C32×C6 C62 C22×C6 C3×C6 C3×C6 C2×C6 C6 # reps 1 1 4 1 1 2 2 8 2 2 4 2 12 8 16 4 24 32

Matrix representation of C6×C327D4 in GL4(𝔽13) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 3 0 0 0 5 9
,
 3 0 0 0 9 9 0 0 0 0 1 0 0 0 0 1
,
 3 2 0 0 9 10 0 0 0 0 8 7 0 0 0 5
,
 10 11 0 0 4 3 0 0 0 0 8 7 0 0 4 5
G:=sub<GL(4,GF(13))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,5,0,0,0,9],[3,9,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[3,9,0,0,2,10,0,0,0,0,8,0,0,0,7,5],[10,4,0,0,11,3,0,0,0,0,8,4,0,0,7,5] >;

C6×C327D4 in GAP, Magma, Sage, TeX

C_6\times C_3^2\rtimes_7D_4
% in TeX

G:=Group("C6xC3^2:7D4");
// GroupNames label

G:=SmallGroup(432,719);
// by ID

G=gap.SmallGroup(432,719);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^4=e^2=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=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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