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G = C22×2+ 1+4  order 128 = 27

Direct product of C22 and 2+ 1+4

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C22×2+ 1+4, D4⋊4C24, C2.4C26, Q8⋊4C24, C4.15C25, C25⋊9C22, C23⋊2C24, C24⋊10C23, C22.18C25, D4○(C22×D4), Q8○(C22×Q8), (C2×C4)⋊2C24, C4○D4⋊7C23, (D4×C23)⋊19C2, (C2×D4)⋊26C23, (C2×Q8)⋊27C23, (C23×C4)⋊53C22, (C22×C4)⋊19C23, (C22×D4)⋊69C22, (C22×Q8)⋊74C22, (C2×D4)○2(C2×D4), (C2×Q8)○2(C2×Q8), (C2×D4)○(C22×D4), (C2×Q8)○(C22×Q8), (C22×D4)○(C22×D4), (C2×C4○D4)⋊79C22, (C22×C4○D4)⋊28C2, (C22×Q8)○(C22×Q8), SmallGroup(128,2323)

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

Derived series
C1 — C2 — C22×2+ 1+4
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C22×2+ 1+4
Lower central
C1 — C2 — C22×2+ 1+4
Upper central
C1 — C23 — C22×2+ 1+4
Jennings
C1 — C2 — C22×2+ 1+4

Generators and relations for C22×2+ 1+4
 G = < a,b,c,d,e,f | a2=b2=c4=d2=f2=1, e2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=c2e >

Subgroups: 4252 in 3280 conjugacy classes, 2836 normal (4 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, Q8, C23, C23, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C24, C23×C4, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, C25, D4×C23, C22×C4○D4, C2×2+ 1+4, C22×2+ 1+4
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C26, C22×2+ 1+4

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

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

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

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

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

68 conjugacy classes

class 1 2A···2G2H···2AQ4A···4X
order12···22···24···4
size11···12···22···2

68 irreducible representations

dim11114
type+++++
imageC1C2C2C22+ 1+4
kernelC22×2+ 1+4D4×C23C22×C4○D4C2×2+ 1+4C22
# reps196484

Matrix representation of C22×2+ 1+4 ►in GL6(ℤ)

100000
0-10000
001000
000100
000010
000001
,
-100000
010000
001000
000100
000010
000001
,
100000
0-10000
0011-1-2
0000-10
000100
0010-1-1
,
100000
0-10000
001000
000100
0000-10
00110-1
,
-100000
010000
000100
00-1000
0011-1-2
00-1011
,
-100000
0-10000
000-100
00-1000
0011-1-2
00-1-101

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,1,0,0,1,0,1,0,0,0,-1,-1,0,-1,0,0,-2,0,0,-1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,-1,0,0,1,0,1,0,0,0,0,0,-1,1,0,0,0,0,-2,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,-1,0,0,-1,0,1,-1,0,0,0,0,-1,0,0,0,0,0,-2,1] >;
 

C22×2+ 1+4 in GAP, Magma, Sage, TeX 

C_2^2\times 2_+^{1+4}
 
% in TeX
 
G:=Group("C2^2xES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(128,2323);
 
// by ID
 
G=gap.SmallGroup(128,2323);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,2,-2,925,723,2019]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=d^2=f^2=1,e^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c^2*e>;
 
// generators/relations
 

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