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G = (C2×C8).168D4order 128 = 27

136th non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Aliases: (C2×C8).168D4, (C2×D4).14Q8, C428C49C2, C2.16(C83D4), C23.939(C2×D4), (C22×C4).164D4, C2.14(D4.Q8), C4.39(C22⋊Q8), C2.18(C8.12D4), C22.129(C4○D8), C22.78(C41D4), (C2×C42).390C22, (C22×C8).330C22, (C22×D4).95C22, C2.5(C23.4Q8), C22.158(C8⋊C22), (C22×C4).1473C23, C22.7C4222C2, C4.33(C22.D4), C22.116(C22⋊Q8), C2.14(C23.19D4), C24.3C22.21C2, C22.126(C22.D4), (C2×C4.Q8)⋊25C2, (C2×C2.D8)⋊13C2, (C2×C4).748(C2×D4), (C2×C4).290(C2×Q8), (C2×D4⋊C4).22C2, (C2×C4).779(C4○D4), (C2×C4⋊C4).162C22, SmallGroup(128,824)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C8).168D4
C1C2C22C2×C4C22×C4C22×D4C24.3C22 — (C2×C8).168D4
C1C2C22×C4 — (C2×C8).168D4
C1C23C2×C42 — (C2×C8).168D4
C1C2C2C22×C4 — (C2×C8).168D4

Generators and relations for (C2×C8).168D4
 G = < a,b,c,d | a2=b8=1, c4=b4, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab5, dbd-1=b3, dcd-1=b4c3 >

Subgroups: 336 in 137 conjugacy classes, 50 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×5], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×6], C2×C4 [×15], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, C2.C42 [×2], D4⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C22×C8 [×2], C22×D4, C22.7C42, C428C4, C24.3C22, C2×D4⋊C4 [×2], C2×C4.Q8, C2×C2.D8, (C2×C8).168D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C4○D8 [×2], C8⋊C22 [×2], C23.4Q8, D4.Q8 [×2], C23.19D4 [×2], C8.12D4, C83D4, (C2×C8).168D4

Smallest permutation representation of (C2×C8).168D4
On 64 points
Generators in S64
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(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 21 29 34 5 17 25 38)(2 51 30 47 6 55 26 43)(3 23 31 36 7 19 27 40)(4 53 32 41 8 49 28 45)(9 35 64 18 13 39 60 22)(10 48 57 56 14 44 61 52)(11 37 58 20 15 33 62 24)(12 42 59 50 16 46 63 54)
(1 39 63 47)(2 34 64 42)(3 37 57 45)(4 40 58 48)(5 35 59 43)(6 38 60 46)(7 33 61 41)(8 36 62 44)(9 50 26 17)(10 53 27 20)(11 56 28 23)(12 51 29 18)(13 54 30 21)(14 49 31 24)(15 52 32 19)(16 55 25 22)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G4H4I···4N8A···8H
order12···222444444444···48···8
size11···188222244448···84···4

32 irreducible representations

dim1111111222224
type+++++++++-+
imageC1C2C2C2C2C2C2D4D4Q8C4○D4C4○D8C8⋊C22
kernel(C2×C8).168D4C22.7C42C428C4C24.3C22C2×D4⋊C4C2×C4.Q8C2×C2.D8C2×C8C22×C4C2×D4C2×C4C22C22
# reps1111211422682

Matrix representation of (C2×C8).168D4 in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
400000
0130000
002100
00141500
0000512
000055
,
040000
1300000
0091300
003800
00001414
0000314
,
010000
1600000
004000
000400
000004
000040

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],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,2,14,0,0,0,0,1,15,0,0,0,0,0,0,5,5,0,0,0,0,12,5],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,9,3,0,0,0,0,13,8,0,0,0,0,0,0,14,3,0,0,0,0,14,14],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0] >;

(C2×C8).168D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{168}D_4
% in TeX

G:=Group("(C2xC8).168D4");
// GroupNames label

G:=SmallGroup(128,824);
// by ID

G=gap.SmallGroup(128,824);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,1018,521,248,1411,718,172]);
// Polycyclic

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

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