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G = C2×C8.17D4order 128 = 27

Direct product of C2 and C8.17D4

direct product, p-group, metabelian, nilpotent (class 4), monomial

Aliases: C2×C8.17D4, C23.37SD16, M5(2).20C22, C8.97(C2×D4), C4.65(C2×D8), (C2×C4).143D8, (C2×C8).124D4, C8.9(C22×C4), (C2×Q16).14C4, Q16.12(C2×C4), (C2×C4).54SD16, C8.11(C22⋊C4), (C2×C8).227C23, (C22×C4).337D4, C4.27(D4⋊C4), (C2×M5(2)).22C2, (C22×Q16).14C2, C22.17(C2×SD16), C8.C4.10C22, (C22×C8).236C22, (C2×Q16).148C22, C22.34(D4⋊C4), (C2×C8).86(C2×C4), (C2×C4).272(C2×D4), C4.59(C2×C22⋊C4), C2.37(C2×D4⋊C4), (C2×C8.C4).21C2, (C2×C4).275(C22⋊C4), SmallGroup(128,879)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C2×C8.17D4
C1C2C4C2×C4C2×C8C22×C8C22×Q16 — C2×C8.17D4
C1C2C4C8 — C2×C8.17D4
C1C22C22×C4C22×C8 — C2×C8.17D4
C1C2C2C2C2C4C4C2×C8 — C2×C8.17D4

Generators and relations for C2×C8.17D4
 G = < a,b,c,d | a2=b8=1, c4=b4, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Subgroups: 212 in 108 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×6], Q8 [×10], C23, C16 [×2], C2×C8 [×6], C2×C8, M4(2) [×3], Q16 [×4], Q16 [×6], C22×C4, C22×C4, C2×Q8 [×9], C8.C4 [×2], C8.C4, C2×C16, M5(2) [×2], M5(2), C22×C8, C2×M4(2), C2×Q16 [×6], C2×Q16 [×3], C22×Q8, C8.17D4 [×4], C2×C8.C4, C2×M5(2), C22×Q16, C2×C8.17D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C8.17D4 [×2], C2×D4⋊C4, C2×C8.17D4

Smallest permutation representation of C2×C8.17D4
On 64 points
Generators in S64
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 49)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 32 30 28 26 24 22 20)(33 47 45 43 41 39 37 35)(34 48 46 44 42 40 38 36)(49 63 61 59 57 55 53 51)(50 64 62 60 58 56 54 52)
(1 19 13 23 9 27 5 31)(2 22 6 18 10 30 14 26)(3 17 15 21 11 25 7 29)(4 20 8 32 12 28 16 24)(33 58 37 54 41 50 45 62)(34 53 46 57 42 61 38 49)(35 56 39 52 43 64 47 60)(36 51 48 55 44 59 40 63)
(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)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J16A···16H
order12222244444444888888888816···16
size1111222222888822224488884···4

32 irreducible representations

dim111111222224
type++++++++-
imageC1C2C2C2C2C4D4D4D8SD16SD16C8.17D4
kernelC2×C8.17D4C8.17D4C2×C8.C4C2×M5(2)C22×Q16C2×Q16C2×C8C22×C4C2×C4C2×C4C23C2
# reps141118314224

Matrix representation of C2×C8.17D4 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
100000
010000
0014300
00141400
00751414
00127314
,
880000
1190000
0061220
0013202
00611115
0085415
,
990000
1080000
0071480
0016909
008121014
00413168

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,14,7,12,0,0,3,14,5,7,0,0,0,0,14,3,0,0,0,0,14,14],[8,11,0,0,0,0,8,9,0,0,0,0,0,0,6,13,6,8,0,0,12,2,11,5,0,0,2,0,11,4,0,0,0,2,5,15],[9,10,0,0,0,0,9,8,0,0,0,0,0,0,7,16,8,4,0,0,14,9,12,13,0,0,8,0,10,16,0,0,0,9,14,8] >;

C2×C8.17D4 in GAP, Magma, Sage, TeX

C_2\times C_8._{17}D_4
% in TeX

G:=Group("C2xC8.17D4");
// GroupNames label

G:=SmallGroup(128,879);
// by ID

G=gap.SmallGroup(128,879);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1123,1466,136,1411,172,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

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