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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, C24.109D4, C8.20(C2×D4), (C2×C8).143D4, C4⋊C4.22C23, C4.Q8⋊49C22, (C2×C8).249C23, (C2×C4).257C24, (C2×Q16)⋊51C22, (C22×Q16)⋊15C2, C4.151(C22×D4), (C22×C4).427D4, C23.863(C2×D4), (C2×Q8).48C23, C4.112(C4⋊D4), Q8⋊C4⋊95C22, (C22×C8).257C22, (C23×C4).549C22, C22.517(C22×D4), (C22×M4(2)).5C2, C22⋊Q8.153C22, C22.176(C4⋊D4), (C22×C4).1536C23, (C22×Q8).280C22, C22.105(C8.C22), (C2×M4(2)).262C22, (C2×C4.Q8)⋊10C2, C4.24(C2×C4○D4), (C2×C4).473(C2×D4), C2.75(C2×C4⋊D4), (C2×Q8⋊C4)⋊55C2, C2.18(C2×C8.C22), (C2×C22⋊Q8).53C2, (C2×C4).703(C4○D4), (C2×C4⋊C4).590C22, SmallGroup(128,1785)

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

Generators and relations for C2×C8.D4
 G = < a,b,c,d | a2=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b3, dbd-1=b-1, dcd-1=b4c-1 >

Subgroups: 436 in 246 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, Q8⋊C4, C4.Q8, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C22⋊Q8, C22×C8, C2×M4(2), C2×M4(2), C2×Q16, C2×Q16, C23×C4, C22×Q8, C2×Q8⋊C4, C2×C4.Q8, C8.D4, C2×C22⋊Q8, C22×M4(2), C22×Q16, C2×C8.D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C8.C22, C22×D4, C2×C4○D4, C8.D4, C2×C4⋊D4, C2×C8.C22, C2×C8.D4

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N8A···8H
order12···2224444444···48···8
size11···1442222448···84···4

32 irreducible representations

dim111111122224
type++++++++++-
imageC1C2C2C2C2C2C2D4D4D4C4○D4C8.C22
kernelC2×C8.D4C2×Q8⋊C4C2×C4.Q8C8.D4C2×C22⋊Q8C22×M4(2)C22×Q16C2×C8C22×C4C24C2×C4C22
# reps121821143144

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

160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
160000000
016000000
00120000
0016160000
000000160
000000016
00000100
000016000
,
72000000
910000000
001080000
001170000
000000716
0000001610
000010100
00001700
,
72000000
1010000000
001080000
001170000
000000101
00000017
000010100
00001700

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

C2×C8.D4 in GAP, Magma, Sage, TeX 

C_2\times C_8.D_4
 
% in TeX
 
G:=Group("C2xC8.D4");
 
// GroupNames label
 
G:=SmallGroup(128,1785);
 
// by ID
 
G=gap.SmallGroup(128,1785);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,723,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,d*b*d^-1=b^-1,d*c*d^-1=b^4*c^-1>;
 
// generators/relations
 

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