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G = C2×C8.2D4  order 128 = 27

Direct product of C2 and C8.2D4

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

Aliases: C2×C8.2D4, C42.246D4, C42.371C23, C8.27(C2×D4), (C2×C8).145D4, C4⋊Q8⋊66C22, C4.7(C22×D4), C8⋊C4⋊45C22, C4.16(C4⋊1D4), (C2×C8).265C23, (C2×C4).347C24, (C22×Q16)⋊18C2, (C2×Q16)⋊55C22, C23.881(C2×D4), (C22×C4).466D4, (C2×D4).113C23, (C2×Q8).101C23, (C22×SD16).5C2, C22.52(C4⋊1D4), (C22×C8).269C22, (C2×C42).853C22, C22.607(C22×D4), (C22×C4).1562C23, (C2×SD16).114C22, C4.4D4.140C22, (C22×D4).375C22, (C22×Q8).308C22, C22.114(C8.C22), (C2×C4⋊Q8)⋊38C2, (C2×C8⋊C4)⋊9C2, (C2×C4).857(C2×D4), C2.26(C2×C4⋊1D4), C2.41(C2×C8.C22), (C2×C4.4D4).41C2, SmallGroup(128,1881)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C2×C8.2D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C8⋊C4 — C2×C8.2D4
Lower central
C1 — C2 — C2×C4 — C2×C8.2D4
Upper central
C1 — C23 — C2×C42 — C2×C8.2D4
Jennings
C1 — C2 — C2 — C2×C4 — C2×C8.2D4

Subgroups: 548 in 282 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×10], C22, C22 [×6], C22 [×10], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×6], Q8 [×18], C23, C23 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×12], SD16 [×16], Q16 [×16], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×6], C2×Q8 [×15], C24, C8⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C2×SD16 [×8], C2×SD16 [×8], C2×Q16 [×8], C2×Q16 [×8], C22×D4, C22×Q8, C22×Q8 [×2], C2×C8⋊C4, C8.2D4 [×8], C2×C4.4D4, C2×C4⋊Q8, C22×SD16 [×2], C22×Q16 [×2], C2×C8.2D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C4⋊1D4 [×4], C8.C22 [×4], C22×D4 [×3], C8.2D4 [×4], C2×C4⋊1D4, C2×C8.C22 [×2], C2×C8.2D4

Generators and relations
 G = < a,b,c,d | a2=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b3, dcd=c3 >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
016000000
00010000
001600000
000021189
000016090
000067161
0000110716
,
11000000
1516000000
000160000
00100000
000039116
00007060
000081710
0000101617
,
10000000
1516000000
001600000
00010000
00001200
000001600
0000016016
000001160

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G4H4I···4N8A···8H
order12···222444444444···48···8
size11···188222244448···84···4

32 irreducible representations

dim11111112224
type++++++++++-
imageC1C2C2C2C2C2C2D4D4D4C8.C22
kernelC2×C8.2D4C2×C8⋊C4C8.2D4C2×C4.4D4C2×C4⋊Q8C22×SD16C22×Q16C42C2×C8C22×C4C22
# reps11811222824

In GAP, Magma, Sage, TeX 

C_2\times C_8._2D_4
 
% in TeX
 
G:=Group("C2xC8.2D4");
 
// GroupNames label
 
G:=SmallGroup(128,1881);
 
// by ID
 
G=gap.SmallGroup(128,1881);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,184,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^3,d*c*d=c^3>;
 
// generators/relations
 

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