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G = C2×C82D4order 128 = 27

Direct product of C2 and C82D4

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

Aliases: C2×C82D4, C24.108D4, (C2×C8)⋊9D4, C85(C2×D4), (C22×D8)⋊15C2, (C2×D8)⋊46C22, C4⋊C4.21C23, C4.Q848C22, C4⋊D454C22, (C2×C8).248C23, (C2×C4).256C24, (C2×D4).60C23, (C22×C4).426D4, C4.150(C22×D4), C23.862(C2×D4), C4.111(C4⋊D4), D4⋊C491C22, (C22×M4(2))⋊2C2, (C2×M4(2))⋊51C22, (C23×C4).548C22, (C22×C8).256C22, C22.516(C22×D4), C22.175(C4⋊D4), C22.116(C8⋊C22), (C22×C4).1535C23, (C22×D4).347C22, (C2×C4.Q8)⋊9C2, C4.23(C2×C4○D4), (C2×C4⋊D4)⋊47C2, (C2×C4).472(C2×D4), C2.74(C2×C4⋊D4), C2.18(C2×C8⋊C22), (C2×D4⋊C4)⋊54C2, (C2×C4).702(C4○D4), (C2×C4⋊C4).589C22, SmallGroup(128,1784)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C82D4
C1C2C22C2×C4C22×C4C23×C4C22×M4(2) — C2×C82D4
C1C2C2×C4 — C2×C82D4
C1C23C23×C4 — C2×C82D4
C1C2C2C2×C4 — C2×C82D4

Generators and relations for C2×C82D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b3, dbd=b-1, dcd=c-1 >

Subgroups: 692 in 298 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×30], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×18], D4 [×28], C23, C23 [×2], C23 [×22], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], C2×C8 [×2], M4(2) [×8], D8 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×4], C2×D4 [×26], C24, C24 [×2], D4⋊C4 [×8], C4.Q8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊D4 [×8], C4⋊D4 [×4], C22×C8 [×2], C2×M4(2) [×4], C2×M4(2) [×4], C2×D8 [×4], C2×D8 [×4], C23×C4, C22×D4 [×2], C22×D4 [×2], C2×D4⋊C4 [×2], C2×C4.Q8, C82D4 [×8], C2×C4⋊D4 [×2], C22×M4(2), C22×D8, C2×C82D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C8⋊C22 [×4], C22×D4 [×2], C2×C4○D4, C82D4 [×4], C2×C4⋊D4, C2×C8⋊C22 [×2], C2×C82D4

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J8A···8H
order12···222222244444444448···8
size11···144888822224488884···4

32 irreducible representations

dim111111122224
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D4C4○D4C8⋊C22
kernelC2×C82D4C2×D4⋊C4C2×C4.Q8C82D4C2×C4⋊D4C22×M4(2)C22×D8C2×C8C22×C4C24C2×C4C22
# reps121821143144

Matrix representation of C2×C82D4 in GL8(𝔽17)

160000000
016000000
00100000
00010000
00001000
00000100
00000010
00000001
,
49000000
013000000
001600000
000160000
000000160
000000016
0000161500
00001100
,
162000000
161000000
00010000
001600000
00001000
0000161600
00000012
000000016
,
10000000
116000000
00100000
000160000
00001000
0000161600
0000001615
00000001

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16],[1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,1] >;

C2×C82D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_2D_4
% in TeX

G:=Group("C2xC8:2D4");
// GroupNames label

G:=SmallGroup(128,1784);
// by ID

G=gap.SmallGroup(128,1784);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,723,2804,172]);
// Polycyclic

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

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