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G = C23.304C24  order 128 = 27

21st central stem extension by C23 of C24

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C25.32C22, C24.242C23, C23.304C24, C22.1222+ (1+4), (C2×D4)⋊41D4, (C22×C4)⋊19D4, C23⋊5(C4○D4), C23⋊2D4⋊4C2, C23⋊Q8⋊3C2, (C23×C4)⋊19C22, C23.147(C2×D4), C2.12(D4⋊5D4), C22.14C22≀C2, (C22×D4)⋊54C22, (C22×Q8)⋊52C22, C23.23D4⋊21C2, C23.34D4⋊17C2, (C22×C4).786C23, C22.184(C22×D4), C2.C42⋊15C22, C2.8(C22.29C24), C2.16(C22.19C24), (C2×C22≀C2)⋊3C2, (C2×C4).300(C2×D4), (C22×C4○D4)⋊2C2, C2.11(C2×C22≀C2), (C2×C22⋊C4)⋊9C22, (C22×C22⋊C4)⋊16C2, C22.183(C2×C4○D4), SmallGroup(128,1136)

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

Derived series
C1 — C23 — C23.304C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C23.304C24
Lower central
C1 — C23 — C23.304C24
Upper central
C1 — C23 — C23.304C24
Jennings
C1 — C23 — C23.304C24

Subgroups: 1140 in 528 conjugacy classes, 120 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×12], C4 [×14], C22, C22 [×10], C22 [×68], C2×C4 [×8], C2×C4 [×54], D4 [×40], Q8 [×8], C23, C23 [×14], C23 [×60], C22⋊C4 [×24], C22×C4 [×2], C22×C4 [×12], C22×C4 [×20], C2×D4 [×8], C2×D4 [×34], C2×Q8 [×6], C4○D4 [×32], C24, C24 [×4], C24 [×12], C2.C42 [×8], C2×C22⋊C4 [×14], C2×C22⋊C4 [×4], C22≀C2 [×8], C23×C4 [×2], C23×C4 [×2], C22×D4, C22×D4 [×6], C22×Q8, C2×C4○D4 [×12], C25, C23.34D4, C23.23D4 [×6], C23⋊2D4 [×2], C23⋊Q8 [×2], C22×C22⋊C4, C2×C22≀C2 [×2], C22×C4○D4, C23.304C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C22≀C2 [×4], C22×D4 [×3], C2×C4○D4 [×2], 2+ (1+4) [×2], C2×C22≀C2, C22.19C24, C22.29C24, D4⋊5D4 [×4], C23.304C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=g2=1, e2=a, ab=ba, ac=ca, ede-1=gdg=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

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

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

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

(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 21)(2 20)(3 23)(4 18)(5 9)(7 11)(13 30)(15 32)(17 27)(19 25)(22 26)(24 28)

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
010000
100000
001300
000400
000004
000040
,
300000
020000
003000
003200
000001
000010
,
400000
010000
001000
000100
000040
000001
,
400000
010000
001000
001400
000010
000001

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,3,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S4A4B4C4D4E···4N4O4P4Q4R
order12···222222···244444···44444
size11···122224···422224···48888

38 irreducible representations

dim111111112224
type+++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D42+ (1+4)
kernelC23.304C24C23.34D4C23.23D4C23⋊2D4C23⋊Q8C22×C22⋊C4C2×C22≀C2C22×C4○D4C22×C4C2×D4C23C22
# reps116221214882

In GAP, Magma, Sage, TeX 

C_2^3._{304}C_2^4
 
% in TeX
 
G:=Group("C2^3.304C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1136);
 
// by ID
 
G=gap.SmallGroup(128,1136);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=g^2=1,e^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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