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G = (C2×C8).169D4order 128 = 27

137th non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Aliases: (C2×C8).169D4, (C2×D4).16Q8, C429C48C2, (C2×C4).43SD16, C2.17(C83D4), C2.14(C85D4), C23.941(C2×D4), (C22×C4).325D4, C4.41(C22⋊Q8), C2.10(D42Q8), C22.80(C41D4), (C22×C8).331C22, (C2×C42).392C22, C22.108(C2×SD16), (C22×D4).97C22, C2.7(C23.4Q8), C22.159(C8⋊C22), (C22×C4).1475C23, C22.7C4232C2, C4.35(C22.D4), C22.118(C22⋊Q8), C2.10(C23.46D4), C24.3C22.23C2, C22.128(C22.D4), (C2×C4.Q8)⋊26C2, (C2×C4).750(C2×D4), (C2×C4).292(C2×Q8), (C2×D4⋊C4).30C2, (C2×C4).781(C4○D4), (C2×C4⋊C4).164C22, SmallGroup(128,826)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C8).169D4
C1C2C22C2×C4C22×C4C22×D4C24.3C22 — (C2×C8).169D4
C1C2C22×C4 — (C2×C8).169D4
C1C23C2×C42 — (C2×C8).169D4
C1C2C2C22×C4 — (C2×C8).169D4

Generators and relations for (C2×C8).169D4
 G = < a,b,c,d | a2=b8=1, c4=b4, d2=ab4, ab=ba, ac=ca, ad=da, cbc-1=ab5, dbd-1=b3, dcd-1=c3 >

Subgroups: 352 in 145 conjugacy classes, 54 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, D4⋊C4 [×4], C4.Q8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22×D4, C22.7C42, C429C4, C24.3C22, C2×D4⋊C4 [×2], C2×C4.Q8 [×2], (C2×C8).169D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, SD16 [×4], C2×D4 [×3], C2×Q8, C4○D4 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C2×SD16 [×2], C8⋊C22 [×2], C23.4Q8, D42Q8 [×2], C23.46D4 [×2], C85D4, C83D4, (C2×C8).169D4

Smallest permutation representation of (C2×C8).169D4
On 64 points
Generators in S64
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(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 20 30 38 5 24 26 34)(2 55 31 47 6 51 27 43)(3 22 32 40 7 18 28 36)(4 49 25 41 8 53 29 45)(9 35 60 21 13 39 64 17)(10 44 61 56 14 48 57 52)(11 37 62 23 15 33 58 19)(12 46 63 50 16 42 59 54)
(1 39 59 47)(2 34 60 42)(3 37 61 45)(4 40 62 48)(5 35 63 43)(6 38 64 46)(7 33 57 41)(8 36 58 44)(9 54 27 20)(10 49 28 23)(11 52 29 18)(12 55 30 21)(13 50 31 24)(14 53 32 19)(15 56 25 22)(16 51 26 17)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111222224
type++++++++-+
imageC1C2C2C2C2C2D4D4Q8SD16C4○D4C8⋊C22
kernel(C2×C8).169D4C22.7C42C429C4C24.3C22C2×D4⋊C4C2×C4.Q8C2×C8C22×C4C2×D4C2×C4C2×C4C22
# reps111122422862

Matrix representation of (C2×C8).169D4 in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
550000
1250000
0051200
005500
000009
000020
,
12120000
5120000
005500
0012500
000002
000090
,
100000
0160000
0001600
0016000
0000130
0000013

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],[5,12,0,0,0,0,5,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,2,0,0,0,0,9,0],[12,5,0,0,0,0,12,12,0,0,0,0,0,0,5,12,0,0,0,0,5,5,0,0,0,0,0,0,0,9,0,0,0,0,2,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13] >;

(C2×C8).169D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{169}D_4
% in TeX

G:=Group("(C2xC8).169D4");
// GroupNames label

G:=SmallGroup(128,826);
// by ID

G=gap.SmallGroup(128,826);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,504,141,624,422,387,1018,521,248,1411,718,172]);
// Polycyclic

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

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