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G = C42.262C23order 128 = 27

123rd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.262C23, C4⋊C884C22, (C4×C8)⋊54C22, C24.78(C2×C4), C8⋊C456C22, C42.206(C2×C4), (C2×C8).475C23, (C2×C4).643C24, C82M4(2)⋊28C2, C42.6C445C2, C42⋊C2.30C4, C22.15(C8○D4), C42.12C447C2, C4.30(C42⋊C2), C2.12(Q8○M4(2)), C22⋊C8.229C22, C24.4C4.24C2, C23.226(C22×C4), (C2×C42).756C22, (C23×C4).523C22, C22.171(C23×C4), (C22×C8).431C22, C42.7C2221C2, C42.6C2229C2, (C22×C4).1273C23, C42⋊C2.349C22, C22.13(C42⋊C2), (C2×M4(2)).345C22, (C2×C4⋊C4).69C4, C2.12(C2×C8○D4), C4⋊C4.219(C2×C4), C4.294(C2×C4○D4), C22⋊C4.89(C2×C4), (C2×C22⋊C8).47C2, (C2×C22⋊C4).46C4, (C2×C4).828(C4○D4), C22⋊C8(C42⋊C2), (C2×C4).259(C22×C4), (C22×C4).337(C2×C4), C2.43(C2×C42⋊C2), (C2×C42⋊C2).58C2, SmallGroup(128,1656)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.262C23
C1C2C4C2×C4C22×C4C42⋊C2C2×C42⋊C2 — C42.262C23
C1C22 — C42.262C23
C1C2×C4 — C42.262C23
C1C2C2C2×C4 — C42.262C23

Generators and relations for C42.262C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, de=ed >

Subgroups: 284 in 201 conjugacy classes, 134 normal (36 characteristic)
C1, C2 [×3], C2 [×5], C4 [×4], C4 [×10], C22, C22 [×4], C22 [×11], C8 [×8], C2×C4 [×4], C2×C4 [×12], C2×C4 [×16], C23 [×3], C23 [×5], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×6], C22×C4 [×4], C22×C4 [×4], C24, C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×2], C22⋊C8 [×6], C4⋊C8 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×4], C22×C8 [×2], C2×M4(2) [×2], C23×C4, C82M4(2) [×2], C2×C22⋊C8, C24.4C4, C42.6C22 [×2], C42.12C4 [×2], C42.6C4 [×2], C42.7C22 [×4], C2×C42⋊C2, C42.262C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C8○D4 [×2], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C2×C8○D4, Q8○M4(2), C42.262C23

Smallest permutation representation of C42.262C23
On 32 points
Generators in S32
(1 19 31 15)(2 16 32 20)(3 21 25 9)(4 10 26 22)(5 23 27 11)(6 12 28 24)(7 17 29 13)(8 14 30 18)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 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)
(2 32)(4 26)(6 28)(8 30)(10 22)(12 24)(14 18)(16 20)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E···4P4Q···4U8A···8H8I···8T
order12222222244444···44···48···88···8
size11112222411112···24···42···24···4

50 irreducible representations

dim111111111111224
type+++++++++
imageC1C2C2C2C2C2C2C2C2C4C4C4C4○D4C8○D4Q8○M4(2)
kernelC42.262C23C82M4(2)C2×C22⋊C8C24.4C4C42.6C22C42.12C4C42.6C4C42.7C22C2×C42⋊C2C2×C22⋊C4C2×C4⋊C4C42⋊C2C2×C4C22C2
# reps121122241448882

Matrix representation of C42.262C23 in GL4(𝔽17) generated by

0100
1000
0040
00913
,
4000
0400
0010
0001
,
2000
0200
0022
00715
,
1000
0100
0010
001516
,
1000
01600
00160
00016
G:=sub<GL(4,GF(17))| [0,1,0,0,1,0,0,0,0,0,4,9,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[2,0,0,0,0,2,0,0,0,0,2,7,0,0,2,15],[1,0,0,0,0,1,0,0,0,0,1,15,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16] >;

C42.262C23 in GAP, Magma, Sage, TeX

C_4^2._{262}C_2^3
% in TeX

G:=Group("C4^2.262C2^3");
// GroupNames label

G:=SmallGroup(128,1656);
// by ID

G=gap.SmallGroup(128,1656);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,521,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,c*e=e*c,d*e=e*d>;
// generators/relations

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