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G = C2×C8⋊3D4  order 128 = 27

Direct product of C2 and C8⋊3D4

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

Aliases: C2×C8⋊3D4, C42.245D4, C42.370C23, C8⋊7(C2×D4), (C2×C8)⋊15D4, C4.6(C22×D4), (C2×D8)⋊50C22, (C22×D8)⋊18C2, C8⋊C4⋊44C22, C4.15(C4⋊1D4), C4⋊1D4⋊37C22, (C2×C8).264C23, (C2×C4).346C24, (C22×SD16)⋊4C2, C23.880(C2×D4), (C22×C4).465D4, (C2×SD16)⋊58C22, (C2×D4).112C23, C4.4D4⋊57C22, (C2×Q8).100C23, C22.51(C4⋊1D4), (C2×C42).852C22, (C22×C8).268C22, C22.606(C22×D4), C22.125(C8⋊C22), (C22×C4).1561C23, (C22×D4).374C22, (C22×Q8).307C22, (C2×C8⋊C4)⋊8C2, (C2×C4⋊1D4)⋊18C2, (C2×C4).856(C2×D4), C2.25(C2×C4⋊1D4), C2.41(C2×C8⋊C22), (C2×C4.4D4)⋊41C2, SmallGroup(128,1880)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C2×C8⋊3D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C8⋊C4 — C2×C8⋊3D4
Lower central
C1 — C2 — C2×C4 — C2×C8⋊3D4
Upper central
C1 — C23 — C2×C42 — C2×C8⋊3D4
Jennings
C1 — C2 — C2 — C2×C4 — C2×C8⋊3D4

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], C4⋊1D4 [×4], C4⋊1D4 [×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, C8⋊3D4 [×8], C2×C4.4D4, C2×C4⋊1D4, C22×D8 [×2], C22×SD16 [×2], C2×C8⋊3D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C4⋊1D4 [×4], C8⋊C22 [×4], C22×D4 [×3], C8⋊3D4 [×4], C2×C4⋊1D4, C2×C8⋊C22 [×2], C2×C8⋊3D4

Generators and relations
 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 >

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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×C8⋊3D4C2×C8⋊C4C8⋊3D4C2×C4.4D4C2×C4⋊1D4C22×D8C22×SD16C42C2×C8C22×C4C22
# reps11811222824

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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