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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), (C4×D4)⋊81C22, (C2×D8)⋊40C22, C4.60(C22×D4), C22.70(C2×D8), C4.63(C4⋊D4), C4⋊1D4⋊32C22, C4⋊C4.370C23, (C2×C4).233C24, (C2×C8).132C23, (C2×D4).45C23, (C22×C4).789D4, C23.852(C2×D4), D4⋊C4⋊63C22, (C22×C8).136C22, (C2×C42).802C22, 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

Subgroups: 780 in 324 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×2], C4 [×6], C4 [×4], C22, C22 [×6], C22 [×36], C8 [×4], C2×C4 [×2], C2×C4 [×12], C2×C4 [×14], D4 [×4], D4 [×34], C23, C23 [×26], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×4], D8 [×16], C22×C4 [×3], C22×C4 [×9], C2×D4 [×10], C2×D4 [×29], C24 [×3], D4⋊C4 [×8], C4⋊C8 [×4], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4 [×4], C4×D4 [×2], C4⋊1D4 [×4], C4⋊1D4 [×2], C22×C8 [×2], C2×D8 [×8], C2×D8 [×8], C23×C4, C22×D4, C22×D4 [×2], C22×D4 [×2], C2×D4⋊C4 [×2], C2×C4⋊C8, C4⋊D8 [×8], C2×C4×D4, C2×C4⋊1D4, C22×D8 [×2], C2×C4⋊D8

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D8 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×D8 [×6], C8⋊C22 [×2], C22×D4 [×2], C2×C4○D4, C4⋊D8 [×4], C2×C4⋊D4, C22×D8, C2×C8⋊C22, C2×C4⋊D8

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

Smallest permutation representation
►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)])
 

Matrix representation ►G ⊆ 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] >;
 

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

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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