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G = C2×C8⋊D4  order 128 = 27

Direct product of C2 and C8⋊D4

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

Aliases: C2×C8⋊D4, C24.107D4, (C2×C8)⋊8D4, C8⋊4(C2×D4), C4⋊C4.20C23, C2.D8⋊68C22, (C2×C4).255C24, (C2×C8).247C23, (C22×SD16)⋊1C2, (C2×D4).59C23, C4.149(C22×D4), C23.861(C2×D4), (C22×C4).425D4, C22⋊Q8⋊66C22, (C2×Q8).47C23, C4.110(C4⋊D4), D4⋊C4⋊90C22, Q8⋊C4⋊94C22, (C2×SD16)⋊54C22, (C22×M4(2))⋊1C2, C4⋊D4.148C22, (C2×M4(2))⋊50C22, (C22×C8).255C22, (C23×C4).547C22, C22.515(C22×D4), C22.174(C4⋊D4), C22.115(C8⋊C22), (C22×C4).1534C23, (C22×D4).346C22, (C22×Q8).279C22, C22.104(C8.C22), (C2×C2.D8)⋊40C2, C4.22(C2×C4○D4), (C2×C4).471(C2×D4), C2.73(C2×C4⋊D4), C2.17(C2×C8⋊C22), (C2×C22⋊Q8)⋊55C2, (C2×D4⋊C4)⋊53C2, (C2×Q8⋊C4)⋊54C2, (C2×C4⋊D4).54C2, C2.17(C2×C8.C22), (C2×C4).701(C4○D4), (C2×C4⋊C4).588C22, SmallGroup(128,1783)

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

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

Subgroups: 564 in 272 conjugacy classes, 108 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×20], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×22], D4 [×14], Q8 [×6], C23, C23 [×2], C23 [×14], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×2], M4(2) [×8], SD16 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×7], C2×D4 [×2], C2×D4 [×13], C2×Q8 [×2], C2×Q8 [×5], C24, C24, D4⋊C4 [×4], Q8⋊C4 [×4], C2.D8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C2×M4(2) [×4], C2×M4(2) [×4], C2×SD16 [×4], C2×SD16 [×4], C23×C4, C22×D4, C22×D4, C22×Q8, C2×D4⋊C4, C2×Q8⋊C4, C2×C2.D8, C8⋊D4 [×8], C2×C4⋊D4, C2×C22⋊Q8, C22×M4(2), C22×SD16, C2×C8⋊D4

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

10000000
01000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
016000000
00100000
00010000
0000143011
0000314116
000031400
0000311146
,
016000000
10000000
000160000
00100000
00000010
0000116162
00001000
000010161
,
160000000
01000000
00100000
000160000
000016000
00000100
000000160
000010161

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,14,3,3,3,0,0,0,0,3,14,14,11,0,0,0,0,0,11,0,14,0,0,0,0,11,6,0,6],[0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,16,0,0,0,0,0,2,0,1],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1] >;
 

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G···4L8A···8H
order12···222224444444···48···8
size11···144882222448···84···4

32 irreducible representations

dim111111111222244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4D4C4○D4C8⋊C22C8.C22
kernelC2×C8⋊D4C2×D4⋊C4C2×Q8⋊C4C2×C2.D8C8⋊D4C2×C4⋊D4C2×C22⋊Q8C22×M4(2)C22×SD16C2×C8C22×C4C24C2×C4C22C22
# reps111181111431422

In GAP, Magma, Sage, TeX 

C_2\times C_8\rtimes D_4
 
% in TeX
 
G:=Group("C2xC8:D4");
 
// GroupNames label
 
G:=SmallGroup(128,1783);
 
// by ID
 
G=gap.SmallGroup(128,1783);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,2804,172]);
 
// 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^-1,d*b*d=b^3,d*c*d=c^-1>;
 
// generators/relations
 

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