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## G = C32×C8⋊C22order 288 = 25·32

### Direct product of C32 and C8⋊C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C32×C8⋊C22
 Chief series C1 — C2 — C4 — C12 — C3×C12 — D4×C32 — C32×D8 — C32×C8⋊C22
 Lower central C1 — C2 — C4 — C32×C8⋊C22
 Upper central C1 — C3×C6 — C6×C12 — C32×C8⋊C22

Generators and relations for C32×C8⋊C22
G = < a,b,c,d,e | a3=b3=c8=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, de=ed >

Subgroups: 348 in 204 conjugacy classes, 120 normal (24 characteristic)
C1, C2, C2 [×4], C3 [×4], C4 [×2], C4, C22, C22 [×5], C6 [×4], C6 [×16], C8 [×2], C2×C4, C2×C4, D4, D4 [×2], D4 [×2], Q8, C23, C32, C12 [×8], C12 [×4], C2×C6 [×4], C2×C6 [×20], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×C6, C3×C6 [×4], C24 [×8], C2×C12 [×4], C2×C12 [×4], C3×D4 [×12], C3×D4 [×8], C3×Q8 [×4], C22×C6 [×4], C8⋊C22, C3×C12 [×2], C3×C12, C62, C62 [×5], C3×M4(2) [×4], C3×D8 [×8], C3×SD16 [×8], C6×D4 [×4], C3×C4○D4 [×4], C3×C24 [×2], C6×C12, C6×C12, D4×C32, D4×C32 [×2], D4×C32 [×2], Q8×C32, C2×C62, C3×C8⋊C22 [×4], C32×M4(2), C32×D8 [×2], C32×SD16 [×2], D4×C3×C6, C32×C4○D4, C32×C8⋊C22
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], D4 [×2], C23, C32, C2×C6 [×28], C2×D4, C3×C6 [×7], C3×D4 [×8], C22×C6 [×4], C8⋊C22, C62 [×7], C6×D4 [×4], D4×C32 [×2], C2×C62, C3×C8⋊C22 [×4], D4×C3×C6, C32×C8⋊C22

Smallest permutation representation of C32×C8⋊C22
On 72 points
Generators in S72
(1 34 59)(2 35 60)(3 36 61)(4 37 62)(5 38 63)(6 39 64)(7 40 57)(8 33 58)(9 68 47)(10 69 48)(11 70 41)(12 71 42)(13 72 43)(14 65 44)(15 66 45)(16 67 46)(17 25 52)(18 26 53)(19 27 54)(20 28 55)(21 29 56)(22 30 49)(23 31 50)(24 32 51)
(1 30 16)(2 31 9)(3 32 10)(4 25 11)(5 26 12)(6 27 13)(7 28 14)(8 29 15)(17 41 62)(18 42 63)(19 43 64)(20 44 57)(21 45 58)(22 46 59)(23 47 60)(24 48 61)(33 56 66)(34 49 67)(35 50 68)(36 51 69)(37 52 70)(38 53 71)(39 54 72)(40 55 65)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(50 52)(51 55)(54 56)(57 61)(58 64)(60 62)(65 69)(66 72)(68 70)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)

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

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

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

99 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3H 4A 4B 4C 6A ··· 6H 6I ··· 6P 6Q ··· 6AN 8A 8B 12A ··· 12P 12Q ··· 12X 24A ··· 24P order 1 2 2 2 2 2 3 ··· 3 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 2 4 4 4 1 ··· 1 2 2 4 1 ··· 1 2 ··· 2 4 ··· 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 C3×D4 C3×D4 C8⋊C22 C3×C8⋊C22 kernel C32×C8⋊C22 C32×M4(2) C32×D8 C32×SD16 D4×C3×C6 C32×C4○D4 C3×C8⋊C22 C3×M4(2) C3×D8 C3×SD16 C6×D4 C3×C4○D4 C3×C12 C62 C12 C2×C6 C32 C3 # reps 1 1 2 2 1 1 8 8 16 16 8 8 1 1 8 8 1 8

Matrix representation of C32×C8⋊C22 in GL6(𝔽73)

 64 0 0 0 0 0 0 64 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 64 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 3 71 0 0 0 0 5 70 0 0 0 0 0 0 15 0 1 39 0 0 56 0 0 19 0 0 72 1 0 57 0 0 71 0 0 58
,
 1 0 0 0 0 0 3 72 0 0 0 0 0 0 1 0 58 15 0 0 0 72 17 0 0 0 0 0 1 0 0 0 0 0 2 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 15 0 0 0 1 0 56 0 0 0 0 72 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,5,0,0,0,0,71,70,0,0,0,0,0,0,15,56,72,71,0,0,0,0,1,0,0,0,1,0,0,0,0,0,39,19,57,58],[1,3,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,58,17,1,2,0,0,15,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,15,56,0,72] >;

C32×C8⋊C22 in GAP, Magma, Sage, TeX

C_3^2\times C_8\rtimes C_2^2
% in TeX

G:=Group("C3^2xC8:C2^2");
// GroupNames label

G:=SmallGroup(288,833);
// by ID

G=gap.SmallGroup(288,833);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,3110,9077,4548,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^2=e^2=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,d*c*d=c^3,e*c*e=c^5,d*e=e*d>;
// generators/relations

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