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G = C2×C23⋊C8  order 128 = 27

Direct product of C2 and C23⋊C8

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

Aliases: C2×C23⋊C8, C24⋊2C8, C25.2C4, C23.27M4(2), C23⋊4(C2×C8), (C23×C4).5C4, C22⋊C8⋊48C22, C24.104(C2×C4), (C22×C4).648D4, C22.6(C22×C8), (C23×C4).15C22, C22.9(C2×M4(2)), C22.43(C23⋊C4), C22.25(C22⋊C8), (C22×C4).420C23, C23.157(C22×C4), C23.163(C22⋊C4), C22.26(C4.D4), (C2×C22⋊C8)⋊1C2, C2.1(C2×C23⋊C4), C2.4(C2×C22⋊C8), C2.1(C2×C4.D4), (C2×C4).1117(C2×D4), (C2×C22⋊C4).15C4, (C22×C4).102(C2×C4), (C22×C22⋊C4).5C2, C22.88(C2×C22⋊C4), (C2×C4).161(C22⋊C4), (C2×C22⋊C4).398C22, SmallGroup(128,188)

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

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

Generators and relations for C2×C23⋊C8
 G = < a,b,c,d,e | a2=b2=c2=d2=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 564 in 228 conjugacy classes, 68 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C24, C24, C24, C22⋊C8, C22⋊C8, C2×C22⋊C4, C2×C22⋊C4, C22×C8, C23×C4, C25, C23⋊C8, C2×C22⋊C8, C22×C22⋊C4, C2×C23⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C23⋊C4, C4.D4, C2×C22⋊C4, C22×C8, C2×M4(2), C23⋊C8, C2×C22⋊C8, C2×C23⋊C4, C2×C4.D4, C2×C23⋊C8

Smallest permutation representation of C2×C23⋊C8
►On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)

(1 11)(3 21)(4 25)(5 15)(7 17)(8 29)(9 28)(10 18)(13 32)(14 22)(19 30)(23 26)

(1 19)(2 12)(3 21)(4 14)(5 23)(6 16)(7 17)(8 10)(9 28)(11 30)(13 32)(15 26)(18 29)(20 31)(22 25)(24 27)

(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)

(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)
 
G:=sub<Sym(32)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,11)(3,21)(4,25)(5,15)(7,17)(8,29)(9,28)(10,18)(13,32)(14,22)(19,30)(23,26), (1,19)(2,12)(3,21)(4,14)(5,23)(6,16)(7,17)(8,10)(9,28)(11,30)(13,32)(15,26)(18,29)(20,31)(22,25)(24,27), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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)>;
 
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,11)(3,21)(4,25)(5,15)(7,17)(8,29)(9,28)(10,18)(13,32)(14,22)(19,30)(23,26), (1,19)(2,12)(3,21)(4,14)(5,23)(6,16)(7,17)(8,10)(9,28)(11,30)(13,32)(15,26)(18,29)(20,31)(22,25)(24,27), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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) );
 
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27)], [(1,11),(3,21),(4,25),(5,15),(7,17),(8,29),(9,28),(10,18),(13,32),(14,22),(19,30),(23,26)], [(1,19),(2,12),(3,21),(4,14),(5,23),(6,16),(7,17),(8,10),(9,28),(11,30),(13,32),(15,26),(18,29),(20,31),(22,25),(24,27)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)], [(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)]])
 

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I4J4K4L8A···8P
order12···2222222224···444448···8
size11···1222244442···244444···4

44 irreducible representations

dim111111112244
type+++++++
imageC1C2C2C2C4C4C4C8D4M4(2)C23⋊C4C4.D4
kernelC2×C23⋊C8C23⋊C8C2×C22⋊C8C22×C22⋊C4C2×C22⋊C4C23×C4C25C24C22×C4C23C22C22
# reps1421422164422

Matrix representation of C2×C23⋊C8 ►in GL8(𝔽17)

160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
01000000
001600000
00910000
000016000
00000100
000000160
00000001
,
160000000
016000000
001600000
000160000
00001000
00000100
000000160
000000016
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
016000000
10000000
008150000
00090000
000000160
000000016
000001600
000016000

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

C2×C23⋊C8 in GAP, Magma, Sage, TeX 

C_2\times C_2^3\rtimes C_8
 
% in TeX
 
G:=Group("C2xC2^3:C8");
 
// GroupNames label
 
G:=SmallGroup(128,188);
 
// by ID
 
G=gap.SmallGroup(128,188);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1123,851,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
 
// generators/relations
 

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