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G = C2xC22:D8  order 128 = 27

Direct product of C2 and C22:D8

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

Aliases: C2xC22:D8, C23:5D8, C24.173D4, D4:6(C2xD4), (C2xD4):47D4, C4:C4:1C23, (C2xC8):3C23, C22:3(C2xD8), (D4xC23):7C2, (C2xD4):1C23, (C22xD8):5C2, C2.4(C22xD8), C4.33C22wrC2, (C2xD8):35C22, (C22xC8):7C22, C4.33(C22xD4), C4:D4:47C22, C22:C8:51C22, (C2xC4).215C24, (C22xC4).415D4, C23.846(C2xD4), D4:C4:61C22, (C22xD4):55C22, C22.112C22wrC2, (C23xC4).535C22, (C22xC4).953C23, C22.475(C22xD4), C22.109(C8:C22), C2.7(C2xC8:C22), (C2xC4:D4):43C2, (C2xC4:C4):44C22, (C2xC22:C8):20C2, (C2xD4:C4):20C2, C2.33(C2xC22wrC2), (C2xC4).1089(C2xD4), SmallGroup(128,1728)

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

Derived series
C1 — C2xC4 — C2xC22:D8
Chief series
C1 — C2 — C22 — C2xC4 — C22xC4 — C23xC4 — D4xC23 — C2xC22:D8
Lower central
C1 — C2 — C2xC4 — C2xC22:D8
Upper central
C1 — C23 — C23xC4 — C2xC22:D8
Jennings
C1 — C2 — C2 — C2xC4 — C2xC22:D8

Generators and relations for C2xC22:D8
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1276 in 524 conjugacy classes, 124 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, C23, C23, C23, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, D8, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C24, C24, C22:C8, D4:C4, C2xC22:C4, C2xC4:C4, C4:D4, C4:D4, C22xC8, C2xD8, C2xD8, C23xC4, C22xD4, C22xD4, C22xD4, C25, C2xC22:C8, C2xD4:C4, C22:D8, C2xC4:D4, C22xD8, D4xC23, C2xC22:D8
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C24, C22wrC2, C2xD8, C8:C22, C22xD4, C22:D8, C2xC22wrC2, C22xD8, C2xC8:C22, C2xC22:D8

Smallest permutation representation of C2xC22:D8
►On 32 points
Generators in S32
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)

(2 29)(4 31)(6 25)(8 27)(9 18)(11 20)(13 22)(15 24)

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

(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)

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S2T2U4A4B4C4D4E4F4G4H8A···8H
order12···222222···222444444448···8
size11···122224···488222244884···4

38 irreducible representations

dim111111122224
type++++++++++++
imageC1C2C2C2C2C2C2D4D4D4D8C8:C22
kernelC2xC22:D8C2xC22:C8C2xD4:C4C22:D8C2xC4:D4C22xD8D4xC23C22xC4C2xD4C24C23C22
# reps112812138182

Matrix representation of C2xC22:D8 ►in GL5(F17)

160000
01000
00100
000160
000016
,
160000
01000
00100
00010
000116
,
10000
01000
00100
000160
000016
,
160000
031400
03300
000115
000116
,
160000
03300
031400
000115
000016

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,3,3,0,0,0,14,3,0,0,0,0,0,1,1,0,0,0,15,16],[16,0,0,0,0,0,3,3,0,0,0,3,14,0,0,0,0,0,1,0,0,0,0,15,16] >;
 

C2xC22:D8 in GAP, Magma, Sage, TeX 

C_2\times C_2^2\rtimes D_8
 
% in TeX
 
G:=Group("C2xC2^2:D8");
 
// GroupNames label
 
G:=SmallGroup(128,1728);
 
// by ID
 
G=gap.SmallGroup(128,1728);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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