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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

Aliases: C32×C8⋊C22, C8⋊C62, D42C62, Q83C62, C62.99D4, C246(C2×C6), (C3×D8)⋊6C6, D82(C3×C6), (C6×D4)⋊14C6, C6.95(C6×D4), (C3×SD16)⋊5C6, SD161(C3×C6), C12.80(C3×D4), C4.5(C2×C62), (C2×C4).6C62, (C3×C24)⋊18C22, (C3×C12).181D4, (C32×D8)⋊10C2, (C3×M4(2))⋊3C6, M4(2)⋊1(C3×C6), C4.14(D4×C32), C12.59(C22×C6), (C32×SD16)⋊9C2, C22.5(D4×C32), (C3×C12).189C23, (C6×C12).275C22, (D4×C32)⋊29C22, (C32×M4(2))⋊5C2, (Q8×C32)⋊26C22, (D4×C3×C6)⋊23C2, C2.15(D4×C3×C6), C4○D44(C3×C6), (C2×D4)⋊5(C3×C6), (C3×C4○D4)⋊11C6, (C3×D4)⋊11(C2×C6), (C3×Q8)⋊12(C2×C6), (C2×C6).34(C3×D4), (C2×C12).76(C2×C6), (C3×C6).312(C2×D4), (C32×C4○D4)⋊12C2, SmallGroup(288,833)

Series: Derived Chief Lower central Upper central

C1C4 — C32×C8⋊C22
C1C2C4C12C3×C12D4×C32C32×D8 — C32×C8⋊C22
C1C2C4 — C32×C8⋊C22
C1C3×C6C6×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 2A2B2C2D2E3A···3H4A4B4C6A···6H6I···6P6Q···6AN8A8B12A···12P12Q···12X24A···24P
order1222223···34446···66···66···68812···1212···1224···24
size1124441···12241···12···24···4442···24···44···4

99 irreducible representations

dim111111111111222244
type+++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4C3×D4C3×D4C8⋊C22C3×C8⋊C22
kernelC32×C8⋊C22C32×M4(2)C32×D8C32×SD16D4×C3×C6C32×C4○D4C3×C8⋊C22C3×M4(2)C3×D8C3×SD16C6×D4C3×C4○D4C3×C12C62C12C2×C6C32C3
# reps11221188161688118818

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

6400000
0640000
008000
000800
000080
000008
,
6400000
0640000
001000
000100
000010
000001
,
3710000
5700000
00150139
00560019
00721057
00710058
,
100000
3720000
00105815
00072170
000010
0000272
,
100000
010000
0010015
0001056
0000720
0000072

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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