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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

Aliases: (C2×C8).41D4, (C2×D4).94D4, (C2×Q8).84D4, C4.11C22≀C2, (C22×Q16)⋊2C2, C4.47(C4⋊D4), (C22×C4).142D4, C23.903(C2×D4), C2.14(C8.2D4), C22.205C22≀C2, C2.29(D4.7D4), C2.17(C8.12D4), C22.104(C4○D8), C22.76(C41D4), C2.20(C232D4), (C22×C8).320C22, (C2×C42).351C22, C2.16(Q8.D4), (C22×SD16).10C2, (C22×D4).67C22, (C22×Q8).56C22, C22.224(C4⋊D4), C23.67C238C2, (C22×C4).1437C23, C22.7C4221C2, C22.119(C8.C22), (C2×Q8⋊C4)⋊34C2, (C2×C4).1028(C2×D4), (C2×D4⋊C4).21C2, (C2×C4).876(C4○D4), (C2×C4⋊C4).110C22, (C2×C4.4D4).11C2, SmallGroup(128,747)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C8).41D4
C1C2C22C2×C4C22×C4C22×D4C2×C4.4D4 — (C2×C8).41D4
C1C2C22×C4 — (C2×C8).41D4
C1C23C2×C42 — (C2×C8).41D4
C1C2C2C22×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···2G2H2I4A4B4C4D4E4F4G4H4I···4N8A···8H
order12···222444444444···48···8
size11···188222244448···84···4

32 irreducible representations

dim111111112222224
type++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4D4C4○D4C4○D8C8.C22
kernel(C2×C8).41D4C22.7C42C23.67C23C2×D4⋊C4C2×Q8⋊C4C2×C4.4D4C22×SD16C22×Q16C2×C8C22×C4C2×D4C2×Q8C2×C4C22C22
# reps111111114224282

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

1600000
0160000
0016000
0001600
0000160
0000016
,
400000
0130000
0001100
003600
000053
00001412
,
0130000
1300000
00101000
0012000
00001412
000053
,
0130000
400000
00101000
0012700
00001412
000053

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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