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## G = (C2×C8).41D4order 128 = 27

### 9th non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C8).41D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C2×C4.4D4 — (C2×C8).41D4
 Lower central C1 — C2 — C22×C4 — (C2×C8).41D4
 Upper central C1 — C23 — C2×C42 — (C2×C8).41D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).41D4

Generators and relations for (C2×C8).41D4
G = < a,b,c,d | a2=b8=1, c4=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=ab5, dbd-1=b-1, dcd-1=c3 >

Subgroups: 440 in 195 conjugacy classes, 54 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×6], C2×C4 [×17], D4 [×6], Q8 [×12], C23, C23 [×8], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×4], C2×Q8 [×10], C24, C2.C42 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4.4D4 [×4], C22×C8 [×2], C2×SD16 [×6], C2×Q16 [×6], C22×D4, C22×Q8 [×2], C22.7C42, C23.67C23, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.4D4, C22×SD16, C22×Q16, (C2×C8).41D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C4○D8 [×2], C8.C22 [×2], C232D4, D4.7D4 [×2], Q8.D4 [×2], C8.12D4, C8.2D4, (C2×C8).41D4

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

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 8A ··· 8H order 1 2 ··· 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 8 8 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 C4○D4 C4○D8 C8.C22 kernel (C2×C8).41D4 C22.7C42 C23.67C23 C2×D4⋊C4 C2×Q8⋊C4 C2×C4.4D4 C22×SD16 C22×Q16 C2×C8 C22×C4 C2×D4 C2×Q8 C2×C4 C22 C22 # reps 1 1 1 1 1 1 1 1 4 2 2 4 2 8 2

Matrix representation of (C2×C8).41D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 11 0 0 0 0 3 6 0 0 0 0 0 0 5 3 0 0 0 0 14 12
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 10 10 0 0 0 0 12 0 0 0 0 0 0 0 14 12 0 0 0 0 5 3
,
 0 13 0 0 0 0 4 0 0 0 0 0 0 0 10 10 0 0 0 0 12 7 0 0 0 0 0 0 14 12 0 0 0 0 5 3

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,3,0,0,0,0,11,6,0,0,0,0,0,0,5,14,0,0,0,0,3,12],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,10,12,0,0,0,0,10,0,0,0,0,0,0,0,14,5,0,0,0,0,12,3],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,10,12,0,0,0,0,10,7,0,0,0,0,0,0,14,5,0,0,0,0,12,3] >;

(C2×C8).41D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{41}D_4
% in TeX

G:=Group("(C2xC8).41D4");
// GroupNames label

G:=SmallGroup(128,747);
// by ID

G=gap.SmallGroup(128,747);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic

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

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