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G = C23.4C24  order 128 = 27

4th non-split extension by C23 of C24 acting via C24/C22=C22

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

Aliases: C23.4C24, Q8○(C23⋊C4), (C2×Q8).225D4, (C22×Q8)⋊11C4, (C2×D4).352C23, C22⋊C4.68C23, C23⋊C4.15C22, Q8.16(C22⋊C4), C22.13(C23×C4), C23.59(C22×C4), C22.26(C22×D4), (C22×C4).272C23, (C2×2- 1+4).3C2, C23.C23⋊17C2, C42⋊C2.76C22, C23.32C23⋊4C2, (C22×Q8).255C22, (C2×C4○D4)⋊10C4, (C2×C4).442(C2×D4), C4.30(C2×C22⋊C4), (C2×D4).221(C2×C4), (C22×C4).37(C2×C4), (C2×Q8).199(C2×C4), (C2×C4).107(C22×C4), (C2×C4○D4).81C22, C2.27(C22×C22⋊C4), SmallGroup(128,1616)

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

Derived series
C1 — C22 — C23.4C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — C2×2- 1+4 — C23.4C24
Lower central
C1 — C2 — C22 — C23.4C24
Upper central
C1 — C2 — C22×Q8 — C23.4C24
Jennings
C1 — C2 — C23 — C23.4C24

Generators and relations for C23.4C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=b, f2=g2=c, ab=ba, dad-1=ac=ca, ae=ea, af=fa, ag=ga, ebe=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, ce=ec, gfg-1=cf=fc, cg=gc, ede=acd, df=fd, dg=gd, ef=fe, eg=ge >

Subgroups: 580 in 358 conjugacy classes, 170 normal (7 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C23⋊C4, C42⋊C2, C4×Q8, C22×Q8, C22×Q8, C2×C4○D4, 2- 1+4, C23.C23, C23.32C23, C2×2- 1+4, C23.4C24
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C23×C4, C22×D4, C22×C22⋊C4, C23.4C24

Smallest permutation representation of C23.4C24
►On 32 points
Generators in S32
(2 28)(4 26)(6 17)(8 19)(10 30)(12 32)(14 22)(16 24)

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

(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(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)

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

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

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

41 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4L4M···4AF
order1222222224···44···4
size1122244442···24···4

41 irreducible representations

dim11111128
type+++++-
imageC1C2C2C2C4C4D4C23.4C24
kernelC23.4C24C23.C23C23.32C23C2×2- 1+4C22×Q8C2×C4○D4C2×Q8C1
# reps112218881

Matrix representation of C23.4C24 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
01000000
10000000
00010000
00100000
00000100
00001000
00000001
00000010
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
00001000
00000100
00000010
00000001
01000000
10000000
00010000
00100000
,
00300000
00020000
20000000
03000000
00000030
00000002
00002000
00000300
,
00100000
00010000
40000000
04000000
00000010
00000001
00004000
00000400
,
03000000
30000000
00020000
00200000
00000300
00003000
00000002
00000020

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0],[0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0] >;
 

C23.4C24 in GAP, Magma, Sage, TeX 

C_2^3._4C_2^4
 
% in TeX
 
G:=Group("C2^3.4C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1616);
 
// by ID
 
G=gap.SmallGroup(128,1616);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,352,521,248,2804,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=b,f^2=g^2=c,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,a*f=f*a,a*g=g*a,e*b*e=b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,g*f*g^-1=c*f=f*c,c*g=g*c,e*d*e=a*c*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e>;
 
// generators/relations
 

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