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G = C24.174C23order 128 = 27

14th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.174C23, C24.7(C2×C4), C22.50(C4×D4), C22⋊C4.119D4, C23.9D49C2, (C22×C4).293D4, C23.128(C2×D4), C22.30C22≀C2, C23.34D47C2, C23.121(C4○D4), C22.48(C4⋊D4), C23.193(C22×C4), (C23×C4).262C22, (C22×D4).27C22, C2.38(C23.23D4), C2.27(C23.C23), C22.28(C22.D4), (C2×C4⋊C4)⋊13C4, (C2×C22⋊C4)⋊5C4, (C2×C23⋊C4).7C2, (C2×C42⋊C2)⋊3C2, (C22×C4).19(C2×C4), (C2×C4).196(C22⋊C4), (C2×C22⋊C4).13C22, C22.274(C2×C22⋊C4), (C2×C22.D4).3C2, SmallGroup(128,631)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.174C23
C1C2C22C23C24C23×C4C2×C42⋊C2 — C24.174C23
C1C2C23 — C24.174C23
C1C22C23×C4 — C24.174C23
C1C2C24 — C24.174C23

Generators and relations for C24.174C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=ca=ac, g2=d, ab=ba, eae-1=faf-1=ad=da, ag=ga, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, fcf-1=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef-1=ae, fg=gf >

Subgroups: 420 in 189 conjugacy classes, 56 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×7], C4 [×13], C22 [×3], C22 [×4], C22 [×15], C2×C4 [×4], C2×C4 [×33], D4 [×4], C23 [×3], C23 [×4], C23 [×7], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×6], C22×C4 [×8], C2×D4 [×6], C24 [×2], C2.C42 [×2], C23⋊C4 [×4], C2×C42, C2×C22⋊C4, C2×C22⋊C4 [×6], C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2 [×4], C22.D4 [×4], C23×C4, C22×D4, C23.9D4 [×2], C23.34D4, C2×C23⋊C4 [×2], C2×C42⋊C2, C2×C22.D4, C24.174C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C23.C23 [×2], C24.174C23

Smallest permutation representation of C24.174C23
On 32 points
Generators in S32
(1 13)(2 14)(3 7)(4 8)(5 11)(6 9)(10 15)(12 16)(17 20)(18 19)(21 22)(23 24)(25 31)(26 30)(27 29)(28 32)
(1 3)(2 4)(5 16)(6 15)(7 13)(8 14)(9 10)(11 12)(17 22)(18 23)(19 24)(20 21)(25 32)(26 29)(27 30)(28 31)
(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 2)(3 4)(5 15)(6 16)(7 8)(9 12)(10 11)(13 14)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(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 18 13 17)(2 20 14 19)(3 23 7 22)(4 21 8 24)(5 32 11 26)(6 27 9 31)(10 28 15 30)(12 29 16 25)
(1 11 2 10)(3 12 4 9)(5 14 15 13)(6 7 16 8)(17 32 19 30)(18 26 20 28)(21 31 23 29)(22 25 24 27)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I2J4A4B4C4D4E···4N4O···4U
order12222···2244444···44···4
size11112···2822224···48···8

32 irreducible representations

dim111111112224
type++++++++
imageC1C2C2C2C2C2C4C4D4D4C4○D4C23.C23
kernelC24.174C23C23.9D4C23.34D4C2×C23⋊C4C2×C42⋊C2C2×C22.D4C2×C22⋊C4C2×C4⋊C4C22⋊C4C22×C4C23C2
# reps121211444444

Matrix representation of C24.174C23 in GL6(𝔽5)

100000
010000
004300
000100
004404
001140
,
400000
040000
001000
000100
000010
000001
,
400000
040000
001000
000100
001240
004304
,
100000
010000
004000
000400
000040
000004
,
200000
030000
001000
004400
004304
000010
,
200000
020000
001230
000011
001201
004404
,
040000
400000
002000
000200
000020
000002

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

C24.174C23 in GAP, Magma, Sage, TeX

C_2^4._{174}C_2^3
% in TeX

G:=Group("C2^4.174C2^3");
// GroupNames label

G:=SmallGroup(128,631);
// by ID

G=gap.SmallGroup(128,631);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,2019,1018,2028]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c,f^2=c*a=a*c,g^2=d,a*b=b*a,e*a*e^-1=f*a*f^-1=a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*c*f^-1=c*d=d*c,c*e=e*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,f*e*f^-1=a*e,f*g=g*f>;
// generators/relations

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