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G = C2×C83D4order 128 = 27

Direct product of C2 and C83D4

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

Aliases: C2×C83D4, C42.245D4, C42.370C23, C87(C2×D4), (C2×C8)⋊15D4, C4.6(C22×D4), (C22×D8)⋊18C2, (C2×D8)⋊50C22, C8⋊C444C22, C4.15(C41D4), C41D437C22, (C2×C8).264C23, (C2×C4).346C24, (C22×SD16)⋊4C2, (C22×C4).465D4, C23.880(C2×D4), (C2×SD16)⋊58C22, (C2×D4).112C23, C4.4D457C22, (C2×Q8).100C23, C22.51(C41D4), (C22×C8).268C22, (C2×C42).852C22, C22.606(C22×D4), C22.125(C8⋊C22), (C22×C4).1561C23, (C22×D4).374C22, (C22×Q8).307C22, (C2×C8⋊C4)⋊8C2, (C2×C41D4)⋊18C2, (C2×C4).856(C2×D4), C2.25(C2×C41D4), C2.41(C2×C8⋊C22), (C2×C4.4D4)⋊41C2, SmallGroup(128,1880)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C83D4
C1C2C22C2×C4C22×C4C2×C42C2×C8⋊C4 — C2×C83D4
C1C2C2×C4 — C2×C83D4
C1C23C2×C42 — C2×C83D4
C1C2C2C2×C4 — C2×C83D4

Generators and relations for C2×C83D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 804 in 334 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×4], C4 [×6], C22, C22 [×6], C22 [×30], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×8], D4 [×34], Q8 [×6], C23, C23 [×24], C42 [×4], C22⋊C4 [×8], C2×C8 [×12], D8 [×16], SD16 [×16], C22×C4, C22×C4 [×2], C22×C4, C2×D4 [×6], C2×D4 [×31], C2×Q8 [×2], C2×Q8 [×5], C24 [×3], C8⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C41D4 [×4], C41D4 [×2], C22×C8 [×2], C2×D8 [×8], C2×D8 [×8], C2×SD16 [×8], C2×SD16 [×8], C22×D4, C22×D4 [×2], C22×D4 [×2], C22×Q8, C2×C8⋊C4, C83D4 [×8], C2×C4.4D4, C2×C41D4, C22×D8 [×2], C22×SD16 [×2], C2×C83D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C8⋊C22 [×4], C22×D4 [×3], C83D4 [×4], C2×C41D4, C2×C8⋊C22 [×2], C2×C83D4

Smallest permutation representation of C2×C83D4
On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)
(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 63 23 26)(2 60 24 31)(3 57 17 28)(4 62 18 25)(5 59 19 30)(6 64 20 27)(7 61 21 32)(8 58 22 29)(9 37 47 49)(10 34 48 54)(11 39 41 51)(12 36 42 56)(13 33 43 53)(14 38 44 50)(15 35 45 55)(16 40 46 52)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 56)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 33)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(41 59)(42 58)(43 57)(44 64)(45 63)(46 62)(47 61)(48 60)

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

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

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

32 conjugacy classes

class 1 2A···2G2H···2M4A4B4C4D4E4F4G4H4I4J8A···8H
order12···22···244444444448···8
size11···18···822224444884···4

32 irreducible representations

dim11111112224
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D4C8⋊C22
kernelC2×C83D4C2×C8⋊C4C83D4C2×C4.4D4C2×C41D4C22×D8C22×SD16C42C2×C8C22×C4C22
# reps11811222824

Matrix representation of C2×C83D4 in GL8(𝔽17)

160000000
016000000
00100000
00010000
00001000
00000100
00000010
00000001
,
161000000
151000000
000160000
00100000
00001315011
00001030
000013602
0000513164
,
116000000
216000000
000160000
00100000
000000160
00001601616
000016000
000011610
,
10000000
216000000
001600000
00010000
000016000
000016100
000000160
00002011

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],[16,15,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,13,1,13,5,0,0,0,0,15,0,6,13,0,0,0,0,0,3,0,16,0,0,0,0,11,0,2,4],[1,2,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,16,1,0,0,0,0,0,0,0,16,0,0,0,0,16,16,0,1,0,0,0,0,0,16,0,0],[1,2,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,1] >;

C2×C83D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_3D_4
% in TeX

G:=Group("C2xC8:3D4");
// GroupNames label

G:=SmallGroup(128,1880);
// by ID

G=gap.SmallGroup(128,1880);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,184,2804,172]);
// 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^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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