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G = C22×C4.D4  order 128 = 27

Direct product of C22 and C4.D4

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

Aliases: C22×C4.D4, C25.5C4, M4(2)⋊10C23, (C2×C4).1C24, C24.34(C2×C4), (D4×C23).14C2, (C22×D4).36C4, C4.131(C22×D4), (C22×C4).775D4, (C2×D4).353C23, C22.14(C23×C4), (C2×M4(2))⋊66C22, (C22×M4(2))⋊16C2, (C22×C4).895C23, C23.220(C22×C4), (C23×C4).509C22, C23.230(C22⋊C4), (C22×D4).548C22, C4.69(C2×C22⋊C4), (C2×D4).222(C2×C4), (C2×C4).1399(C2×D4), (C2×C4).242(C22×C4), (C22×C4).321(C2×C4), C22.78(C2×C22⋊C4), C2.28(C22×C22⋊C4), (C2×C4).279(C22⋊C4), SmallGroup(128,1617)

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

Derived series
C1 — C22 — C22×C4.D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — D4×C23 — C22×C4.D4
Lower central
C1 — C2 — C22 — C22×C4.D4
Upper central
C1 — C23 — C23×C4 — C22×C4.D4
Jennings
C1 — C2 — C2 — C2×C4 — C22×C4.D4

Generators and relations for C22×C4.D4
 G = < a,b,c,d,e | a2=b2=c4=1, d4=c2, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=c-1d3 >

Subgroups: 1068 in 496 conjugacy classes, 180 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C23, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C24, C24, C24, C4.D4, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C22×D4, C22×D4, C25, C2×C4.D4, C22×M4(2), D4×C23, C22×C4.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C4.D4, C2×C22⋊C4, C23×C4, C22×D4, C2×C4.D4, C22×C22⋊C4, C22×C4.D4

Smallest permutation representation of C22×C4.D4
►On 32 points
Generators in S32
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S4A···4H8A···8P
order12···222222···24···48···8
size11···122224···42···24···4

44 irreducible representations

dim11111124
type++++++
imageC1C2C2C2C4C4D4C4.D4
kernelC22×C4.D4C2×C4.D4C22×M4(2)D4×C23C22×D4C25C22×C4C22
# reps1122112484

Matrix representation of C22×C4.D4 ►in GL8(𝔽17)

160000000
016000000
00100000
00010000
000016000
000001600
000000160
000000016
,
10000000
01000000
001600000
000160000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
0000161500
00001100
000001301
000044160
,
115000000
116000000
00040000
00400000
0000130015
00000011
000010013
00000104
,
162000000
01000000
00040000
001300000
000040150
00000011
000001130
00001040

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

C22×C4.D4 in GAP, Magma, Sage, TeX 

C_2^2\times C_4.D_4
 
% in TeX
 
G:=Group("C2^2xC4.D4");
 
// GroupNames label
 
G:=SmallGroup(128,1617);
 
// by ID
 
G=gap.SmallGroup(128,1617);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,2804,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^4=c^2,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=c^-1*d^3>;
 
// generators/relations
 

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