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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 [×3], C2 [×4], C2 [×2], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×4], C23, C23 [×8], C42, C22⋊C4 [×4], C4⋊C4, C2×C8 [×8], C22×C4, C22×C4 [×2], C22×C4, C2×D4 [×2], C2×D4 [×3], C24, C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C22×C8 [×2], C22×D4, C22.7C42 [×2], C24.3C22, (C2×C4).98D8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), D8 [×2], SD16 [×2], C22⋊C8, C23⋊C4, C4.D4, D4⋊C4 [×2], C4≀C2 [×2], C23⋊C8, D4⋊C8 [×2], C22.SD16 [×2], C42.C22, C4.D8, (C2×C4).98D8

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

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

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

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

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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