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## G = C2×C33⋊7D4order 432 = 24·33

### Direct product of C2 and C33⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C2×C33⋊7D4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C33⋊7D4 — C2×C33⋊7D4
 Lower central C33 — C32×C6 — C2×C33⋊7D4
 Upper central C1 — C22

Generators and relations for C2×C337D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf=b-1, cd=dc, ece-1=fcf=c-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 3000 in 452 conjugacy classes, 92 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C2×C3⋊S3, C62, C62, C62, C2×D12, C2×C3⋊D4, S3×C32, C33⋊C2, C32×C6, C32×C6, C3⋊D12, C6×Dic3, C2×C3⋊Dic3, C327D4, S3×C2×C6, C22×C3⋊S3, C2×C62, C3×C3⋊Dic3, S3×C3×C6, S3×C3×C6, C2×C33⋊C2, C2×C33⋊C2, C3×C62, C2×C3⋊D12, C2×C327D4, C337D4, C6×C3⋊Dic3, S3×C62, C22×C33⋊C2, C2×C337D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C3⋊D4, C22×S3, S32, C2×C3⋊S3, C2×D12, C2×C3⋊D4, C3⋊D12, C327D4, C2×S32, C22×C3⋊S3, S3×C3⋊S3, C2×C3⋊D12, C2×C327D4, C337D4, C2×S3×C3⋊S3, C2×C337D4

Smallest permutation representation of C2×C337D4
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 29 62)(2 63 30)(3 31 64)(4 61 32)(5 52 27)(6 28 49)(7 50 25)(8 26 51)(9 56 33)(10 34 53)(11 54 35)(12 36 55)(13 20 67)(14 68 17)(15 18 65)(16 66 19)(21 48 72)(22 69 45)(23 46 70)(24 71 47)(37 42 59)(38 60 43)(39 44 57)(40 58 41)
(1 26 15)(2 16 27)(3 28 13)(4 14 25)(5 63 66)(6 67 64)(7 61 68)(8 65 62)(9 58 69)(10 70 59)(11 60 71)(12 72 57)(17 50 32)(18 29 51)(19 52 30)(20 31 49)(21 39 36)(22 33 40)(23 37 34)(24 35 38)(41 45 56)(42 53 46)(43 47 54)(44 55 48)
(1 15 26)(2 16 27)(3 13 28)(4 14 25)(5 63 66)(6 64 67)(7 61 68)(8 62 65)(9 69 58)(10 70 59)(11 71 60)(12 72 57)(17 50 32)(18 51 29)(19 52 30)(20 49 31)(21 39 36)(22 40 33)(23 37 34)(24 38 35)(41 56 45)(42 53 46)(43 54 47)(44 55 48)
(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 38)(2 37)(3 40)(4 39)(5 10)(6 9)(7 12)(8 11)(13 22)(14 21)(15 24)(16 23)(17 48)(18 47)(19 46)(20 45)(25 36)(26 35)(27 34)(28 33)(29 43)(30 42)(31 41)(32 44)(49 56)(50 55)(51 54)(52 53)(57 61)(58 64)(59 63)(60 62)(65 71)(66 70)(67 69)(68 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), (1,29,62)(2,63,30)(3,31,64)(4,61,32)(5,52,27)(6,28,49)(7,50,25)(8,26,51)(9,56,33)(10,34,53)(11,54,35)(12,36,55)(13,20,67)(14,68,17)(15,18,65)(16,66,19)(21,48,72)(22,69,45)(23,46,70)(24,71,47)(37,42,59)(38,60,43)(39,44,57)(40,58,41), (1,26,15)(2,16,27)(3,28,13)(4,14,25)(5,63,66)(6,67,64)(7,61,68)(8,65,62)(9,58,69)(10,70,59)(11,60,71)(12,72,57)(17,50,32)(18,29,51)(19,52,30)(20,31,49)(21,39,36)(22,33,40)(23,37,34)(24,35,38)(41,45,56)(42,53,46)(43,47,54)(44,55,48), (1,15,26)(2,16,27)(3,13,28)(4,14,25)(5,63,66)(6,64,67)(7,61,68)(8,62,65)(9,69,58)(10,70,59)(11,71,60)(12,72,57)(17,50,32)(18,51,29)(19,52,30)(20,49,31)(21,39,36)(22,40,33)(23,37,34)(24,38,35)(41,56,45)(42,53,46)(43,54,47)(44,55,48), (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), (1,38)(2,37)(3,40)(4,39)(5,10)(6,9)(7,12)(8,11)(13,22)(14,21)(15,24)(16,23)(17,48)(18,47)(19,46)(20,45)(25,36)(26,35)(27,34)(28,33)(29,43)(30,42)(31,41)(32,44)(49,56)(50,55)(51,54)(52,53)(57,61)(58,64)(59,63)(60,62)(65,71)(66,70)(67,69)(68,72)>;

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3E 3F 3G 3H 3I 4A 4B 6A ··· 6O 6P ··· 6AA 6AB ··· 6AQ 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 ··· 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 1 1 6 6 54 54 2 ··· 2 4 4 4 4 18 18 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D12 C3⋊D4 S32 C3⋊D12 C2×S32 kernel C2×C33⋊7D4 C33⋊7D4 C6×C3⋊Dic3 S3×C62 C22×C33⋊C2 C2×C3⋊Dic3 S3×C2×C6 C32×C6 C3⋊Dic3 S3×C6 C62 C3×C6 C3×C6 C2×C6 C6 C6 # reps 1 4 1 1 1 1 4 2 2 8 5 4 16 4 8 4

Matrix representation of C2×C337D4 in GL8(ℤ)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C2×C337D4 in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes_7D_4
% in TeX

G:=Group("C2xC3^3:7D4");
// GroupNames label

G:=SmallGroup(432,681);
// by ID

G=gap.SmallGroup(432,681);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,571,2028,14118]);
// Polycyclic

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

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