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G = C2×C23.C23order 128 = 27

Direct product of C2 and C23.C23

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

Aliases: C2×C23.C23, C23.2C24, C24.469C23, (C23×C4)⋊11C4, (C22×D4)⋊25C4, C24.33(C2×C4), (C22×Q8)⋊19C4, C23⋊C411C22, C23.636(C2×D4), (C22×C4).774D4, (C2×D4).350C23, C22⋊C4.66C23, C22.11(C23×C4), C23.57(C22×C4), C22.24(C22×D4), C42⋊C272C22, C4(C23.C23), (C23×C4).508C22, (C22×C4).894C23, (C22×D4).547C22, C4(C2×C23⋊C4), (C2×C4○D4)⋊17C4, (C2×D4)⋊44(C2×C4), (C2×C4)(C23⋊C4), (C22×C4)⋊8(C2×C4), (C2×Q8)⋊35(C2×C4), (C2×C23⋊C4)⋊19C2, C4.68(C2×C22⋊C4), (C2×C4).1398(C2×D4), (C2×C42⋊C2)⋊39C2, (C2×C4).105(C22×C4), (C22×C4○D4).16C2, C2.25(C22×C22⋊C4), (C2×C4).278(C22⋊C4), (C2×C4○D4).270C22, C22.135(C2×C22⋊C4), (C2×C22⋊C4).475C22, SmallGroup(128,1614)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C23.C23
C1C2C22C23C24C23×C4C22×C4○D4 — C2×C23.C23
C1C2C22 — C2×C23.C23
C1C2×C4C23×C4 — C2×C23.C23
C1C2C23 — C2×C23.C23

Generators and relations for C2×C23.C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=c, g2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ebe-1=bd=db, bf=fb, bg=gb, fcf=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef=bde, eg=ge, fg=gf >

Subgroups: 700 in 394 conjugacy classes, 172 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C4 [×12], C22 [×3], C22 [×4], C22 [×26], C2×C4 [×2], C2×C4 [×30], C2×C4 [×36], D4 [×24], Q8 [×8], C23 [×3], C23 [×8], C23 [×10], C42 [×8], C22⋊C4 [×8], C22⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×22], C22×C4 [×12], C2×D4 [×12], C2×D4 [×12], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×2], C23⋊C4 [×16], C2×C42 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C42⋊C2 [×8], C42⋊C2 [×4], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C22×Q8, C2×C4○D4 [×8], C2×C4○D4 [×8], C2×C23⋊C4 [×4], C23.C23 [×8], C2×C42⋊C2 [×2], C22×C4○D4, C2×C23.C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C23.C23 [×2], C22×C22⋊C4, C2×C23.C23

Smallest permutation representation of C2×C23.C23
On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 10)(6 11)(7 12)(8 9)(13 29)(14 30)(15 31)(16 32)(17 24)(18 21)(19 22)(20 23)
(1 6)(2 28)(3 8)(4 26)(5 10)(7 12)(9 25)(11 27)(13 17)(14 30)(15 19)(16 32)(18 21)(20 23)(22 31)(24 29)
(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 11)(2 12)(3 9)(4 10)(5 26)(6 27)(7 28)(8 25)(13 24)(14 21)(15 22)(16 23)(17 29)(18 30)(19 31)(20 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 17)(2 21)(3 31)(4 16)(5 20)(6 13)(7 30)(8 22)(9 19)(10 23)(11 29)(12 14)(15 25)(18 28)(24 27)(26 32)
(1 31 11 19)(2 32 12 20)(3 29 9 17)(4 30 10 18)(5 21 26 14)(6 22 27 15)(7 23 28 16)(8 24 25 13)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A4B4C4D4E···4J4K···4AD
order12222···2222244444···44···4
size11112···2444411112···24···4

44 irreducible representations

dim11111111124
type++++++
imageC1C2C2C2C2C4C4C4C4D4C23.C23
kernelC2×C23.C23C2×C23⋊C4C23.C23C2×C42⋊C2C22×C4○D4C23×C4C22×D4C22×Q8C2×C4○D4C22×C4C2
# reps14821422884

Matrix representation of C2×C23.C23 in GL6(𝔽5)

400000
040000
004000
000400
000040
000004
,
400000
040000
001000
000100
000040
000004
,
400000
040000
001300
000400
000013
000004
,
100000
010000
004000
000400
000040
000004
,
040000
100000
000010
000001
001300
000400
,
400000
010000
002100
002300
000021
000023
,
100000
010000
002000
000200
000020
000002

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

C2×C23.C23 in GAP, Magma, Sage, TeX

C_2\times C_2^3.C_2^3
% in TeX

G:=Group("C2xC2^3.C2^3");
// GroupNames label

G:=SmallGroup(128,1614);
// by ID

G=gap.SmallGroup(128,1614);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,352,2804,2028]);
// Polycyclic

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

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