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G = C2×C4⋊D8  order 128 = 27

Direct product of C2 and C4⋊D8

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

Aliases: C2×C4⋊D8, C42.205D4, C42.314C23, C4⋊3(C2×D8), D4⋊1(C2×D4), (C2×C4)⋊10D8, (C2×D4)⋊37D4, C4⋊C8⋊57C22, (C22×D8)⋊8C2, C2.5(C22×D8), (C2×D8)⋊40C22, (C4×D4)⋊81C22, C22.70(C2×D8), C4.60(C22×D4), C4.63(C4⋊D4), C4⋊1D4⋊32C22, C4⋊C4.370C23, (C2×C8).132C23, (C2×C4).233C24, (C2×D4).45C23, C23.852(C2×D4), (C22×C4).789D4, D4⋊C4⋊63C22, (C2×C42).802C22, (C22×C8).136C22, C22.493(C22×D4), C22.165(C4⋊D4), C22.112(C8⋊C22), (C22×C4).1523C23, (C22×D4).334C22, (C2×C4×D4)⋊59C2, (C2×C4⋊C8)⋊22C2, (C2×C4⋊1D4)⋊15C2, C4.143(C2×C4○D4), C2.51(C2×C4⋊D4), C2.12(C2×C8⋊C22), (C2×D4⋊C4)⋊22C2, (C2×C4).1414(C2×D4), (C2×C4).900(C4○D4), (C2×C4⋊C4).914C22, SmallGroup(128,1761)

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

Derived series
C1 — C2×C4 — C2×C4⋊D8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C4×D4 — C2×C4⋊D8
Lower central
C1 — C2 — C2×C4 — C2×C4⋊D8
Upper central
C1 — C23 — C2×C42 — C2×C4⋊D8
Jennings
C1 — C2 — C2 — C2×C4 — C2×C4⋊D8

Generators and relations for C2×C4⋊D8
 G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 780 in 324 conjugacy classes, 116 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C4⋊1D4, C4⋊1D4, C22×C8, C2×D8, C2×D8, C23×C4, C22×D4, C22×D4, C22×D4, C2×D4⋊C4, C2×C4⋊C8, C4⋊D8, C2×C4×D4, C2×C4⋊1D4, C22×D8, C2×C4⋊D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C24, C4⋊D4, C2×D8, C8⋊C22, C22×D4, C2×C4○D4, C4⋊D8, C2×C4⋊D4, C22×D8, C2×C8⋊C22, C2×C4⋊D8

Smallest permutation representation of C2×C4⋊D8
►On 64 points
Generators in S64
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)

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

(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)

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I···4N8A···8H
order12···2222222224···44···48···8
size11···1444488882···24···44···4

38 irreducible representations

dim1111111222224
type++++++++++++
imageC1C2C2C2C2C2C2D4D4D4D8C4○D4C8⋊C22
kernelC2×C4⋊D8C2×D4⋊C4C2×C4⋊C8C4⋊D8C2×C4×D4C2×C4⋊1D4C22×D8C42C22×C4C2×D4C2×C4C2×C4C22
# reps1218112224842

Matrix representation of C2×C4⋊D8 ►in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
1600000
0160000
001000
000100
0000013
0000130
,
010000
1600000
006600
0014000
0000016
000010
,
100000
0160000
001000
00161600
000010
0000016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,6,14,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
 

C2×C4⋊D8 in GAP, Magma, Sage, TeX 

C_2\times C_4\rtimes D_8
 
% in TeX
 
G:=Group("C2xC4:D8");
 
// GroupNames label
 
G:=SmallGroup(128,1761);
 
// by ID
 
G=gap.SmallGroup(128,1761);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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