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## G = C2×C4×C32⋊C6order 432 = 24·33

### Direct product of C2×C4 and C32⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C4×C32⋊C6
 Chief series C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — C22×C32⋊C6 — C2×C4×C32⋊C6
 Lower central C32 — C2×C4×C32⋊C6
 Upper central C1 — C2×C4

Generators and relations for C2×C4×C32⋊C6
G = < a,b,c,d,e | a2=b4=c3=d3=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 817 in 205 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, C62, S3×C2×C4, C22×C12, C32⋊C6, C2×He3, C2×He3, S3×C12, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C6×C12, S3×C2×C6, C22×C3⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6, C22×He3, S3×C2×C12, C2×C4×C3⋊S3, C4×C32⋊C6, C2×C32⋊C12, C2×C4×He3, C22×C32⋊C6, C2×C4×C32⋊C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, S3×C6, S3×C2×C4, C22×C12, C32⋊C6, S3×C12, S3×C2×C6, C2×C32⋊C6, S3×C2×C12, C4×C32⋊C6, C22×C32⋊C6, C2×C4×C32⋊C6

Smallest permutation representation of C2×C4×C32⋊C6
On 72 points
Generators in S72
(1 21)(2 22)(3 14)(4 13)(5 7)(6 8)(9 20)(10 19)(11 24)(12 23)(15 17)(16 18)(25 56)(26 57)(27 58)(28 59)(29 60)(30 55)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 63)(38 64)(39 65)(40 66)(41 61)(42 62)(43 54)(44 49)(45 50)(46 51)(47 52)(48 53)
(1 11 6 20)(2 12 5 19)(3 18 4 17)(7 10 22 23)(8 9 21 24)(13 15 14 16)(25 45 70 66)(26 46 71 61)(27 47 72 62)(28 48 67 63)(29 43 68 64)(30 44 69 65)(31 37 59 53)(32 38 60 54)(33 39 55 49)(34 40 56 50)(35 41 57 51)(36 42 58 52)
(1 56 35)(2 32 59)(3 58 33)(4 36 55)(5 60 31)(6 34 57)(7 29 67)(8 70 26)(9 66 46)(10 43 63)(11 50 41)(12 38 53)(13 72 30)(14 27 69)(15 62 44)(16 47 65)(17 42 49)(18 52 39)(19 54 37)(20 40 51)(21 25 71)(22 68 28)(23 64 48)(24 45 61)
(1 3 5)(2 6 4)(7 21 14)(8 13 22)(9 15 23)(10 24 16)(11 18 19)(12 20 17)(25 27 29)(26 30 28)(31 35 33)(32 34 36)(37 41 39)(38 40 42)(43 45 47)(44 48 46)(49 53 51)(50 52 54)(55 59 57)(56 58 60)(61 65 63)(62 64 66)(67 71 69)(68 70 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)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D ··· 6I 6J ··· 6R 6S ··· 6Z 12A 12B 12C 12D 12E ··· 12L 12M ··· 12X 12Y ··· 12AF order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 ··· 6 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 9 9 9 9 2 3 3 6 6 6 1 1 1 1 9 9 9 9 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9 2 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 S3 D6 D6 C3×S3 C4×S3 S3×C6 S3×C6 S3×C12 C32⋊C6 C2×C32⋊C6 C2×C32⋊C6 C4×C32⋊C6 kernel C2×C4×C32⋊C6 C4×C32⋊C6 C2×C32⋊C12 C2×C4×He3 C22×C32⋊C6 C2×C4×C3⋊S3 C2×C32⋊C6 C4×C3⋊S3 C2×C3⋊Dic3 C6×C12 C22×C3⋊S3 C2×C3⋊S3 C6×C12 C3×C12 C62 C2×C12 C3×C6 C12 C2×C6 C6 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 2 8 8 2 2 2 16 1 2 1 2 4 4 2 8 1 2 1 4

Matrix representation of C2×C4×C32⋊C6 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0
,
 4 0 0 0 0 0 0 0 9 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0

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

C2×C4×C32⋊C6 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_3^2\rtimes C_6
% in TeX

G:=Group("C2xC4xC3^2:C6");
// GroupNames label

G:=SmallGroup(432,349);
// by ID

G=gap.SmallGroup(432,349);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,142,4037,1034,14118]);
// Polycyclic

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

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