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G = C2.(C82D4)  order 128 = 27

3rd central extension by C2 of C82D4

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

Aliases: C4.125(C4×D4), (C2×C8).200D4, D4⋊C411C4, C2.3(C82D4), C2.5(C8⋊D4), C23.787(C2×D4), C22.170(C4×D4), (C22×C4).131D4, C2.10(D8⋊C4), C22.4Q1649C2, C4.4(C422C2), C4.10(C42⋊C2), C22.88(C8⋊C22), (C22×C8).398C22, (C2×C42).303C22, (C22×D4).40C22, C2.17(SD16⋊C4), C22.133(C4⋊D4), (C22×C4).1387C23, C23.65C234C2, C4.95(C22.D4), C22.61(C4.4D4), C22.77(C8.C22), C24.3C22.9C2, C2.2(C42.29C22), C2.3(C42.28C22), C2.21(C24.C22), C4⋊C4.84(C2×C4), (C2×C8⋊C4)⋊24C2, (C2×C8).147(C2×C4), (C2×D4).100(C2×C4), (C2×C4).1344(C2×D4), (C2×C4⋊C4).73C22, (C2×D4⋊C4).34C2, (C2×C4).582(C4○D4), (C2×C4).405(C22×C4), SmallGroup(128,668)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2.(C82D4)
C1C2C22C2×C4C22×C4C22×C8C2×C8⋊C4 — C2.(C82D4)
C1C2C2×C4 — C2.(C82D4)
C1C23C2×C42 — C2.(C82D4)
C1C2C2C22×C4 — C2.(C82D4)

Generators and relations for C2.(C82D4)
 G = < a,b,c,d | a2=b8=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab3, dbd-1=ab-1, dcd-1=b6c-1 >

Subgroups: 340 in 142 conjugacy classes, 56 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×6], C22 [×7], C22 [×10], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, C2.C42, C8⋊C4 [×2], D4⋊C4 [×4], D4⋊C4 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×D4, C22.4Q16 [×2], C23.65C23, C24.3C22, C2×C8⋊C4, C2×D4⋊C4 [×2], C2.(C82D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C8⋊C22 [×3], C8.C22, C24.C22, SD16⋊C4, D8⋊C4, C8⋊D4, C82D4, C42.28C22, C42.29C22, C2.(C82D4)

Smallest permutation representation of C2.(C82D4)
On 64 points
Generators in S64
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(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 52 16 33)(2 20 9 44)(3 50 10 39)(4 18 11 42)(5 56 12 37)(6 24 13 48)(7 54 14 35)(8 22 15 46)(17 63 41 31)(19 61 43 29)(21 59 45 27)(23 57 47 25)(26 53 58 34)(28 51 60 40)(30 49 62 38)(32 55 64 36)
(1 8 31 30)(2 29 32 7)(3 6 25 28)(4 27 26 5)(9 61 64 14)(10 13 57 60)(11 59 58 12)(15 63 62 16)(17 40 52 48)(18 47 53 39)(19 38 54 46)(20 45 55 37)(21 36 56 44)(22 43 49 35)(23 34 50 42)(24 41 51 33)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111122244
type+++++++++-
imageC1C2C2C2C2C2C4D4D4C4○D4C8⋊C22C8.C22
kernelC2.(C82D4)C22.4Q16C23.65C23C24.3C22C2×C8⋊C4C2×D4⋊C4D4⋊C4C2×C8C22×C4C2×C4C22C22
# reps121112822831

Matrix representation of C2.(C82D4) in GL8(𝔽17)

10000000
01000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
016000000
001390000
00440000
000088140
000098014
000001499
00003089
,
01000000
160000000
001300000
00440000
000014098
00000388
000098014
000088140
,
160000000
01000000
00480000
000130000
000088140
00008903
000014098
00000388

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,4,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,0,8,9,0,3,0,0,0,0,8,8,14,0,0,0,0,0,14,0,9,8,0,0,0,0,0,14,9,9],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,14,0,9,8,0,0,0,0,0,3,8,8,0,0,0,0,9,8,0,14,0,0,0,0,8,8,14,0],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,8,13,0,0,0,0,0,0,0,0,8,8,14,0,0,0,0,0,8,9,0,3,0,0,0,0,14,0,9,8,0,0,0,0,0,3,8,8] >;

C2.(C82D4) in GAP, Magma, Sage, TeX

C_2.(C_8\rtimes_2D_4)
% in TeX

G:=Group("C2.(C8:2D4)");
// GroupNames label

G:=SmallGroup(128,668);
// by ID

G=gap.SmallGroup(128,668);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,723,58,2019,248,2804,172]);
// Polycyclic

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

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