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G = C2×C85D4order 128 = 27

Direct product of C2 and C85D4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×C85D4, C42.356D4, C42.710C23, (C2×C8)⋊32D4, C811(C2×D4), C41(C2×SD16), (C4×C8)⋊77C22, (C2×C4)⋊10SD16, C4⋊Q865C22, C4.1(C22×D4), C4.11(C41D4), (C2×C4).341C24, (C2×C8).592C23, (C22×C4).611D4, C23.876(C2×D4), (C2×Q8).96C23, (C22×SD16)⋊26C2, (C2×SD16)⋊77C22, (C2×D4).108C23, C22.88(C2×SD16), C2.18(C22×SD16), C41D4.150C22, C22.47(C41D4), (C22×C8).566C22, C22.601(C22×D4), (C2×C42).1125C22, (C22×C4).1556C23, (C22×D4).371C22, (C22×Q8).304C22, (C2×C4×C8)⋊35C2, (C2×C4⋊Q8)⋊37C2, (C2×C4).851(C2×D4), C2.20(C2×C41D4), (C2×C41D4).25C2, SmallGroup(128,1875)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C85D4
C1C2C22C2×C4C22×C4C2×C42C2×C4×C8 — C2×C85D4
C1C2C2×C4 — C2×C85D4
C1C23C2×C42 — C2×C85D4
C1C2C2C2×C4 — C2×C85D4

Generators and relations for C2×C85D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b3, dcd=c-1 >

Subgroups: 724 in 328 conjugacy classes, 132 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×12], C4 [×4], C22, C22 [×6], C22 [×20], C8 [×8], C2×C4 [×2], C2×C4 [×16], C2×C4 [×8], D4 [×28], Q8 [×12], C23, C23 [×16], C42 [×4], C4⋊C4 [×8], C2×C8 [×12], SD16 [×32], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C2×Q8 [×4], C2×Q8 [×10], C24 [×2], C4×C8 [×4], C2×C42, C2×C4⋊C4 [×2], C41D4 [×4], C41D4 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C2×SD16 [×16], C2×SD16 [×16], C22×D4 [×2], C22×D4 [×2], C22×Q8 [×2], C2×C4×C8, C85D4 [×8], C2×C41D4, C2×C4⋊Q8, C22×SD16 [×4], C2×C85D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], SD16 [×8], C2×D4 [×18], C24, C41D4 [×4], C2×SD16 [×12], C22×D4 [×3], C85D4 [×4], C2×C41D4, C22×SD16 [×2], C2×C85D4

Smallest permutation representation of C2×C85D4
On 64 points
Generators in S64
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)
(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)
(1 34 63 13)(2 35 64 14)(3 36 57 15)(4 37 58 16)(5 38 59 9)(6 39 60 10)(7 40 61 11)(8 33 62 12)(17 26 56 42)(18 27 49 43)(19 28 50 44)(20 29 51 45)(21 30 52 46)(22 31 53 47)(23 32 54 48)(24 25 55 41)
(1 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)(17 48)(18 43)(19 46)(20 41)(21 44)(22 47)(23 42)(24 45)(25 51)(26 54)(27 49)(28 52)(29 55)(30 50)(31 53)(32 56)(33 60)(34 63)(35 58)(36 61)(37 64)(38 59)(39 62)(40 57)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M4N4O4P8A···8P
order12···222224···444448···8
size11···188882···288882···2

44 irreducible representations

dim1111112222
type+++++++++
imageC1C2C2C2C2C2D4D4D4SD16
kernelC2×C85D4C2×C4×C8C85D4C2×C41D4C2×C4⋊Q8C22×SD16C42C2×C8C22×C4C2×C4
# reps11811428216

Matrix representation of C2×C85D4 in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
010000
1600000
001000
000100
0000125
00001212
,
1600000
0160000
0011500
0011600
000010
000001
,
1600000
010000
0011500
0001600
000010
0000016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C2×C85D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_5D_4
% in TeX

G:=Group("C2xC8:5D4");
// GroupNames label

G:=SmallGroup(128,1875);
// by ID

G=gap.SmallGroup(128,1875);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,184,2804,172]);
// Polycyclic

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

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