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## G = C22×2+ 1+4order 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, D44C24, C2.4C26, Q84C24, C4.15C25, C259C22, C232C24, C2410C23, C22.18C25, D4(C22×D4), Q8(C22×Q8), (C2×C4)⋊2C24, C4○D47C23, (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 ··· 2G 2H ··· 2AQ 4A ··· 4X order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 4 type + + + + + image C1 C2 C2 C2 2+ 1+4 kernel C22×2+ 1+4 D4×C23 C22×C4○D4 C2×2+ 1+4 C22 # reps 1 9 6 48 4

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

 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 1 -1 -2 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 1 0 -1 -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 1 1 0 -1
,
 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 1 1 -1 -2 0 0 -1 0 1 1
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 1 1 -1 -2 0 0 -1 -1 0 1

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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