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## G = (C2×C8).195D4order 128 = 27

### 163rd non-split extension by C2×C8 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — (C2×C8).195D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — (C2×C8).195D4
 Lower central C1 — C23 — (C2×C8).195D4
 Upper central C1 — C22×C4 — (C2×C8).195D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).195D4

Generators and relations for (C2×C8).195D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=b2, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=ab5, dcd-1=b6c3 >

Subgroups: 276 in 162 conjugacy classes, 68 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×2], C4 [×2], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×6], C2×C4 [×2], C2×C4 [×10], C2×C4 [×20], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×4], C2×C8 [×4], C2×C8 [×10], M4(2) [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C2.C42 [×2], C22⋊C8 [×2], C4⋊C8 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×6], C23×C4, C22.7C42 [×2], C4×C22⋊C4, C2×C22⋊C8, C2×C4⋊C8 [×2], C22×M4(2), (C2×C8).195D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C2×M4(2) [×2], C8○D4 [×2], C23.8Q8, C4⋊M4(2), C42.6C22, C89D4 [×2], C86D4 [×2], (C2×C8).195D4

Smallest permutation representation of (C2×C8).195D4
On 64 points
Generators in S64
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)
(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 56 11 19 5 52 15 23)(2 33 12 64 6 37 16 60)(3 50 13 21 7 54 9 17)(4 35 14 58 8 39 10 62)(18 42 51 26 22 46 55 30)(20 44 53 28 24 48 49 32)(25 57 45 38 29 61 41 34)(27 59 47 40 31 63 43 36)
(1 17 3 19 5 21 7 23)(2 58 4 60 6 62 8 64)(9 56 11 50 13 52 15 54)(10 37 12 39 14 33 16 35)(18 46 20 48 22 42 24 44)(25 36 27 38 29 40 31 34)(26 49 28 51 30 53 32 55)(41 63 43 57 45 59 47 61)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4R 8A ··· 8P order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 4 1 ··· 1 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 D4 D4 Q8 M4(2) C4○D4 C8○D4 kernel (C2×C8).195D4 C22.7C42 C4×C22⋊C4 C2×C22⋊C8 C2×C4⋊C8 C22×M4(2) C2.C42 C2×C22⋊C4 C2×C8 C22×C4 C22×C4 C2×C4 C2×C4 C22 # reps 1 2 1 1 2 1 4 4 4 2 2 8 4 8

Matrix representation of (C2×C8).195D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 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 16 0 0 0 0 0 0 0 16 0 0 0 0 4 0 0 0 0 0 0 0 1 2 0 0 0 0 10 16
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 13 0 0 0 0 0 0 0 13 9 0 0 0 0 11 4
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 4 0 0 0 0 0 0 0 13 9 0 0 0 0 10 4

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,16,0,0,0,0,0,0,0,1,10,0,0,0,0,2,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,16,0,0,0,0,0,0,0,13,11,0,0,0,0,9,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,16,0,0,0,0,0,0,0,13,10,0,0,0,0,9,4] >;

(C2×C8).195D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{195}D_4
% in TeX

G:=Group("(C2xC8).195D4");
// GroupNames label

G:=SmallGroup(128,583);
// by ID

G=gap.SmallGroup(128,583);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=b^2,d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^5,d*c*d^-1=b^6*c^3>;
// generators/relations

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