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## G = C32×C23⋊C4order 288 = 25·32

### Direct product of C32 and C23⋊C4

direct product, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×C23⋊C4
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C2×C62 — C32×C22⋊C4 — C32×C23⋊C4
 Lower central C1 — C2 — C22 — C32×C23⋊C4
 Upper central C1 — C3×C6 — C2×C62 — C32×C23⋊C4

Generators and relations for C32×C23⋊C4
G = < a,b,c,d,e,f | a3=b3=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 300 in 156 conjugacy classes, 72 normal (16 characteristic)
C1, C2, C2 [×4], C3 [×4], C4 [×3], C22, C22 [×2], C22 [×3], C6 [×4], C6 [×16], C2×C4, C2×C4 [×2], D4 [×2], C23 [×2], C32, C12 [×12], C2×C6 [×12], C2×C6 [×12], C22⋊C4 [×2], C2×D4, C3×C6, C3×C6 [×4], C2×C12 [×4], C2×C12 [×8], C3×D4 [×8], C22×C6 [×8], C23⋊C4, C3×C12 [×3], C62, C62 [×2], C62 [×3], C3×C22⋊C4 [×8], C6×D4 [×4], C6×C12, C6×C12 [×2], D4×C32 [×2], C2×C62 [×2], C3×C23⋊C4 [×4], C32×C22⋊C4 [×2], D4×C3×C6, C32×C23⋊C4
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, D4 [×2], C32, C12 [×8], C2×C6 [×4], C22⋊C4, C3×C6 [×3], C2×C12 [×4], C3×D4 [×8], C23⋊C4, C3×C12 [×2], C62, C3×C22⋊C4 [×4], C6×C12, D4×C32 [×2], C3×C23⋊C4 [×4], C32×C22⋊C4, C32×C23⋊C4

Smallest permutation representation of C32×C23⋊C4
On 72 points
Generators in S72
(1 5 11)(2 6 12)(3 36 20)(4 35 19)(7 17 9)(8 18 10)(13 22 15)(14 21 16)(23 25 27)(24 26 28)(29 31 33)(30 32 34)(37 70 62)(38 71 63)(39 72 64)(40 69 61)(41 54 49)(42 55 50)(43 56 51)(44 53 52)(45 60 66)(46 57 67)(47 58 68)(48 59 65)
(1 7 22)(2 8 21)(3 30 26)(4 29 25)(5 17 15)(6 18 16)(9 13 11)(10 14 12)(19 33 23)(20 34 24)(27 35 31)(28 36 32)(37 54 60)(38 55 57)(39 56 58)(40 53 59)(41 45 62)(42 46 63)(43 47 64)(44 48 61)(49 66 70)(50 67 71)(51 68 72)(52 65 69)
(1 38)(2 37)(3 39)(4 40)(5 71)(6 70)(7 55)(8 54)(9 42)(10 41)(11 63)(12 62)(13 46)(14 45)(15 67)(16 66)(17 50)(18 49)(19 61)(20 64)(21 60)(22 57)(23 48)(24 47)(25 59)(26 58)(27 65)(28 68)(29 53)(30 56)(31 52)(32 51)(33 44)(34 43)(35 69)(36 72)
(2 3)(6 36)(8 30)(10 34)(12 20)(14 24)(16 28)(18 32)(21 26)(37 39)(41 43)(45 47)(49 51)(54 56)(58 60)(62 64)(66 68)(70 72)
(1 4)(2 3)(5 35)(6 36)(7 29)(8 30)(9 33)(10 34)(11 19)(12 20)(13 23)(14 24)(15 27)(16 28)(17 31)(18 32)(21 26)(22 25)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(65 67)(66 68)(69 71)(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)| (1,5,11)(2,6,12)(3,36,20)(4,35,19)(7,17,9)(8,18,10)(13,22,15)(14,21,16)(23,25,27)(24,26,28)(29,31,33)(30,32,34)(37,70,62)(38,71,63)(39,72,64)(40,69,61)(41,54,49)(42,55,50)(43,56,51)(44,53,52)(45,60,66)(46,57,67)(47,58,68)(48,59,65), (1,7,22)(2,8,21)(3,30,26)(4,29,25)(5,17,15)(6,18,16)(9,13,11)(10,14,12)(19,33,23)(20,34,24)(27,35,31)(28,36,32)(37,54,60)(38,55,57)(39,56,58)(40,53,59)(41,45,62)(42,46,63)(43,47,64)(44,48,61)(49,66,70)(50,67,71)(51,68,72)(52,65,69), (1,38)(2,37)(3,39)(4,40)(5,71)(6,70)(7,55)(8,54)(9,42)(10,41)(11,63)(12,62)(13,46)(14,45)(15,67)(16,66)(17,50)(18,49)(19,61)(20,64)(21,60)(22,57)(23,48)(24,47)(25,59)(26,58)(27,65)(28,68)(29,53)(30,56)(31,52)(32,51)(33,44)(34,43)(35,69)(36,72), (2,3)(6,36)(8,30)(10,34)(12,20)(14,24)(16,28)(18,32)(21,26)(37,39)(41,43)(45,47)(49,51)(54,56)(58,60)(62,64)(66,68)(70,72), (1,4)(2,3)(5,35)(6,36)(7,29)(8,30)(9,33)(10,34)(11,19)(12,20)(13,23)(14,24)(15,27)(16,28)(17,31)(18,32)(21,26)(22,25)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(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( (1,5,11)(2,6,12)(3,36,20)(4,35,19)(7,17,9)(8,18,10)(13,22,15)(14,21,16)(23,25,27)(24,26,28)(29,31,33)(30,32,34)(37,70,62)(38,71,63)(39,72,64)(40,69,61)(41,54,49)(42,55,50)(43,56,51)(44,53,52)(45,60,66)(46,57,67)(47,58,68)(48,59,65), (1,7,22)(2,8,21)(3,30,26)(4,29,25)(5,17,15)(6,18,16)(9,13,11)(10,14,12)(19,33,23)(20,34,24)(27,35,31)(28,36,32)(37,54,60)(38,55,57)(39,56,58)(40,53,59)(41,45,62)(42,46,63)(43,47,64)(44,48,61)(49,66,70)(50,67,71)(51,68,72)(52,65,69), (1,38)(2,37)(3,39)(4,40)(5,71)(6,70)(7,55)(8,54)(9,42)(10,41)(11,63)(12,62)(13,46)(14,45)(15,67)(16,66)(17,50)(18,49)(19,61)(20,64)(21,60)(22,57)(23,48)(24,47)(25,59)(26,58)(27,65)(28,68)(29,53)(30,56)(31,52)(32,51)(33,44)(34,43)(35,69)(36,72), (2,3)(6,36)(8,30)(10,34)(12,20)(14,24)(16,28)(18,32)(21,26)(37,39)(41,43)(45,47)(49,51)(54,56)(58,60)(62,64)(66,68)(70,72), (1,4)(2,3)(5,35)(6,36)(7,29)(8,30)(9,33)(10,34)(11,19)(12,20)(13,23)(14,24)(15,27)(16,28)(17,31)(18,32)(21,26)(22,25)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(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([(1,5,11),(2,6,12),(3,36,20),(4,35,19),(7,17,9),(8,18,10),(13,22,15),(14,21,16),(23,25,27),(24,26,28),(29,31,33),(30,32,34),(37,70,62),(38,71,63),(39,72,64),(40,69,61),(41,54,49),(42,55,50),(43,56,51),(44,53,52),(45,60,66),(46,57,67),(47,58,68),(48,59,65)], [(1,7,22),(2,8,21),(3,30,26),(4,29,25),(5,17,15),(6,18,16),(9,13,11),(10,14,12),(19,33,23),(20,34,24),(27,35,31),(28,36,32),(37,54,60),(38,55,57),(39,56,58),(40,53,59),(41,45,62),(42,46,63),(43,47,64),(44,48,61),(49,66,70),(50,67,71),(51,68,72),(52,65,69)], [(1,38),(2,37),(3,39),(4,40),(5,71),(6,70),(7,55),(8,54),(9,42),(10,41),(11,63),(12,62),(13,46),(14,45),(15,67),(16,66),(17,50),(18,49),(19,61),(20,64),(21,60),(22,57),(23,48),(24,47),(25,59),(26,58),(27,65),(28,68),(29,53),(30,56),(31,52),(32,51),(33,44),(34,43),(35,69),(36,72)], [(2,3),(6,36),(8,30),(10,34),(12,20),(14,24),(16,28),(18,32),(21,26),(37,39),(41,43),(45,47),(49,51),(54,56),(58,60),(62,64),(66,68),(70,72)], [(1,4),(2,3),(5,35),(6,36),(7,29),(8,30),(9,33),(10,34),(11,19),(12,20),(13,23),(14,24),(15,27),(16,28),(17,31),(18,32),(21,26),(22,25),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64),(65,67),(66,68),(69,71),(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)])

99 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3H 4A ··· 4E 6A ··· 6H 6I ··· 6AF 6AG ··· 6AN 12A ··· 12AN order 1 2 2 2 2 2 3 ··· 3 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 2 2 2 4 1 ··· 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 C3×D4 C23⋊C4 C3×C23⋊C4 kernel C32×C23⋊C4 C32×C22⋊C4 D4×C3×C6 C3×C23⋊C4 C6×C12 C2×C62 C3×C22⋊C4 C6×D4 C2×C12 C22×C6 C62 C2×C6 C32 C3 # reps 1 2 1 8 2 2 16 8 16 16 2 16 1 8

Matrix representation of C32×C23⋊C4 in GL6(𝔽13)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 3
,
 3 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 3
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 12 0 0

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

C32×C23⋊C4 in GAP, Magma, Sage, TeX

C_3^2\times C_2^3\rtimes C_4
% in TeX

G:=Group("C3^2xC2^3:C4");
// GroupNames label

G:=SmallGroup(288,317);
// by ID

G=gap.SmallGroup(288,317);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,6304,4548]);
// Polycyclic

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

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