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G = (C2×C4).98D8order 128 = 27

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

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

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 ··· 2G 2H 2I 4A ··· 4H 4I 4J 4K 4L 8A ··· 8P order 1 2 ··· 2 2 2 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 8 8 2 ··· 2 4 4 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C4 C4 C8 D4 M4(2) D8 SD16 C4≀C2 C23⋊C4 C4.D4 kernel (C2×C4).98D8 C22.7C42 C24.3C22 C2×C4⋊C4 C22×D4 C2×D4 C22×C4 C2×C4 C2×C4 C2×C4 C22 C22 C22 # reps 1 2 1 2 2 8 2 2 4 4 8 1 1

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

 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 4 0 0 0 0 3 15
,
 3 3 0 0 0 0 14 3 0 0 0 0 0 0 8 2 0 0 0 0 0 9 0 0 0 0 0 0 15 3 0 0 0 0 4 2
,
 3 3 0 0 0 0 3 14 0 0 0 0 0 0 8 2 0 0 0 0 4 9 0 0 0 0 0 0 15 3 0 0 0 0 15 12

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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