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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

Aliases: C2×C8⋊2D4, C24.108D4, (C2×C8)⋊9D4, C8⋊5(C2×D4), (C22×D8)⋊15C2, (C2×D8)⋊46C22, C4⋊C4.21C23, C4.Q8⋊48C22, C4⋊D4⋊54C22, (C2×C8).248C23, (C2×C4).256C24, (C2×D4).60C23, (C22×C4).426D4, C4.150(C22×D4), C23.862(C2×D4), C4.111(C4⋊D4), D4⋊C4⋊91C22, (C22×M4(2))⋊2C2, (C2×M4(2))⋊51C22, (C23×C4).548C22, (C22×C8).256C22, C22.516(C22×D4), C22.175(C4⋊D4), C22.116(C8⋊C22), (C22×C4).1535C23, (C22×D4).347C22, (C2×C4.Q8)⋊9C2, C4.23(C2×C4○D4), (C2×C4⋊D4)⋊47C2, (C2×C4).472(C2×D4), C2.74(C2×C4⋊D4), C2.18(C2×C8⋊C22), (C2×D4⋊C4)⋊54C2, (C2×C4).702(C4○D4), (C2×C4⋊C4).589C22, SmallGroup(128,1784)

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 — C23×C4 — C22×M4(2) — C2×C8⋊2D4
Lower central
C1 — C2 — C2×C4 — C2×C8⋊2D4
Upper central
C1 — C23 — C23×C4 — C2×C8⋊2D4
Jennings
C1 — C2 — C2 — C2×C4 — C2×C8⋊2D4

Generators and relations for C2×C8⋊2D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b3, dbd=b-1, dcd=c-1 >

Subgroups: 692 in 298 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, D4⋊C4, C4.Q8, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22×C8, C2×M4(2), C2×M4(2), C2×D8, C2×D8, C23×C4, C22×D4, C22×D4, C2×D4⋊C4, C2×C4.Q8, C8⋊2D4, C2×C4⋊D4, C22×M4(2), C22×D8, C2×C8⋊2D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C8⋊C22, C22×D4, C2×C4○D4, C8⋊2D4, C2×C4⋊D4, C2×C8⋊C22, C2×C8⋊2D4

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J8A···8H
order12···222222244444444448···8
size11···144888822224488884···4

32 irreducible representations

dim111111122224
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D4C4○D4C8⋊C22
kernelC2×C8⋊2D4C2×D4⋊C4C2×C4.Q8C8⋊2D4C2×C4⋊D4C22×M4(2)C22×D8C2×C8C22×C4C24C2×C4C22
# reps121821143144

Matrix representation of C2×C8⋊2D4 ►in GL8(𝔽17)

160000000
016000000
00100000
00010000
00001000
00000100
00000010
00000001
,
49000000
013000000
001600000
000160000
000000160
000000016
0000161500
00001100
,
162000000
161000000
00010000
001600000
00001000
0000161600
00000012
000000016
,
10000000
116000000
00100000
000160000
00001000
0000161600
0000001615
00000001

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16],[1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,1] >;
 

C2×C8⋊2D4 in GAP, Magma, Sage, TeX 

C_2\times C_8\rtimes_2D_4
 
% in TeX
 
G:=Group("C2xC8:2D4");
 
// GroupNames label
 
G:=SmallGroup(128,1784);
 
// by ID
 
G=gap.SmallGroup(128,1784);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,723,2804,172]);
 
// 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,c*b*c^-1=b^3,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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