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G = (C2×C4).98D8  order 128 = 27

1st non-split extension by C2×C4 of D8 acting via D8/D4=C2

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

Aliases: (C2×D4)⋊1C8, (C2×C4).98D8, C2.3(D4⋊C8), C22.28C4≀C2, C2.3(C23⋊C8), (C22×D4).1C4, (C2×C4).6M4(2), (C2×C4).109SD16, C2.1(C4.D8), (C22×C4).631D4, C22.30(C23⋊C4), C22.45(C22⋊C8), (C2×C42).118C22, C22.7C42⋊1C2, C2.1(C22.SD16), C22.35(D4⋊C4), C23.209(C22⋊C4), C22.21(C4.D4), C24.3C22.1C2, C2.1(C42.C22), (C2×C4⋊C4).1C4, (C2×C4).6(C2×C8), (C22×C4).142(C2×C4), SmallGroup(128,2)

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

Derived series
C1 — C2×C4 — (C2×C4).98D8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C24.3C22 — (C2×C4).98D8
Lower central
C1 — C22 — C2×C4 — (C2×C4).98D8
Upper central
C1 — C23 — C2×C42 — (C2×C4).98D8
Jennings
C1 — C22 — C23 — C2×C42 — (C2×C4).98D8

Generators and relations for (C2×C4).98D8
 G = < a,b,c,d | a2=b4=c8=1, d2=b-1, cbc-1=ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c-1 >

Subgroups: 280 in 101 conjugacy classes, 34 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.7C42, C24.3C22, (C2×C4).98D8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), D8, SD16, C22⋊C8, C23⋊C4, C4.D4, D4⋊C4, C4≀C2, C23⋊C8, D4⋊C8, C22.SD16, C42.C22, C4.D8, (C2×C4).98D8

Smallest permutation representation of (C2×C4).98D8
►On 64 points
Generators in S64
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(33 55)(34 56)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I4J4K4L8A···8P
order12···2224···444448···8
size11···1882···244884···4

38 irreducible representations

dim1111112222244
type+++++++
imageC1C2C2C4C4C8D4M4(2)D8SD16C4≀C2C23⋊C4C4.D4
kernel(C2×C4).98D8C22.7C42C24.3C22C2×C4⋊C4C22×D4C2×D4C22×C4C2×C4C2×C4C2×C4C22C22C22
# reps1212282244811

Matrix representation of (C2×C4).98D8 ►in GL6(𝔽17)

100000
010000
001000
000100
0000160
0000016
,
100000
010000
0013000
0001300
000024
0000315
,
330000
1430000
008200
000900
0000153
000042
,
330000
3140000
008200
004900
0000153
00001512

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,2,3,0,0,0,0,4,15],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,8,0,0,0,0,0,2,9,0,0,0,0,0,0,15,4,0,0,0,0,3,2],[3,3,0,0,0,0,3,14,0,0,0,0,0,0,8,4,0,0,0,0,2,9,0,0,0,0,0,0,15,15,0,0,0,0,3,12] >;
 

(C2×C4).98D8 in GAP, Magma, Sage, TeX 

(C_2\times C_4)._{98}D_8
 
% in TeX
 
G:=Group("(C2xC4).98D8");
 
// GroupNames label
 
G:=SmallGroup(128,2);
 
// by ID
 
G=gap.SmallGroup(128,2);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,2,56,85,422,387,184,794,248]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=b^-1,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
 
// generators/relations
 

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