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G = C42.677C23order 128 = 27

92nd non-split extension by C42 of C23 acting via C23/C22=C2

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

Aliases: C42.677C23, C4⋊C883C22, (C4×C8)⋊52C22, (C2×C4)⋊12M4(2), (C23×C4).42C4, (C2×C42).57C4, C8⋊C454C22, (C4×M4(2))⋊29C2, (C2×C4).639C24, C24.130(C2×C4), (C2×C8).397C23, C42.334(C2×C4), C4.54(C2×M4(2)), C42(C4⋊M4(2)), C4⋊M4(2)⋊41C2, C42(C24.4C4), C42(C42.6C4), C42.6C458C2, (C22×C42).34C2, C42.12C445C2, C2.9(C22×M4(2)), C4.65(C42⋊C2), C22⋊C8.227C22, C42(C4⋊M4(2)), C24.4C4.25C2, C42(C42.6C4), C42(C24.4C4), C22.167(C23×C4), (C23×C4).698C22, C23.225(C22×C4), C22.28(C2×M4(2)), (C2×C42).1106C22, (C22×C4).1272C23, C22.37(C42⋊C2), (C2×M4(2)).341C22, C4.290(C2×C4○D4), (C2×C4).677(C4○D4), (C2×C4).498(C22×C4), (C22×C4).498(C2×C4), C2.39(C2×C42⋊C2), SmallGroup(128,1652)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.677C23
C1C2C4C2×C4C22×C4C2×C42C22×C42 — C42.677C23
C1C22 — C42.677C23
C1C42 — C42.677C23
C1C2C2C2×C4 — C42.677C23

Generators and relations for C42.677C23
 G = < a,b,c,d,e | a4=b4=1, c2=b, d2=e2=a2, ab=ba, cac-1=a-1b2, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, de=ed >

Subgroups: 332 in 242 conjugacy classes, 152 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C4 [×6], C22, C22 [×6], C22 [×14], C8 [×8], C2×C4 [×2], C2×C4 [×22], C2×C4 [×30], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×12], C2×C8 [×8], M4(2) [×8], C22×C4 [×2], C22×C4 [×16], C22×C4 [×12], C24, C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42 [×2], C2×C42 [×10], C2×M4(2) [×4], C23×C4, C23×C4 [×2], C4×M4(2) [×2], C24.4C4 [×2], C4⋊M4(2) [×2], C42.12C4 [×4], C42.6C4 [×4], C22×C42, C42.677C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×8], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C2×M4(2) [×12], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C22×M4(2) [×2], C42.677C23

Smallest permutation representation of C42.677C23
On 32 points
Generators in S32
(1 10 31 23)(2 20 32 15)(3 12 25 17)(4 22 26 9)(5 14 27 19)(6 24 28 11)(7 16 29 21)(8 18 30 13)
(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)
(1 19 31 14)(2 20 32 15)(3 21 25 16)(4 22 26 9)(5 23 27 10)(6 24 28 11)(7 17 29 12)(8 18 30 13)
(1 19 31 14)(2 15 32 20)(3 21 25 16)(4 9 26 22)(5 23 27 10)(6 11 28 24)(7 17 29 12)(8 13 30 18)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D···2I4A···4L4M···4AD8A···8P
order12222···24···44···48···8
size11112···21···12···24···4

56 irreducible representations

dim11111111122
type+++++++
imageC1C2C2C2C2C2C2C4C4M4(2)C4○D4
kernelC42.677C23C4×M4(2)C24.4C4C4⋊M4(2)C42.12C4C42.6C4C22×C42C2×C42C23×C4C2×C4C2×C4
# reps1222441124168

Matrix representation of C42.677C23 in GL4(𝔽17) generated by

13000
01300
00130
0004
,
13000
01300
0010
0001
,
0100
13000
0001
0010
,
13000
01300
0040
0004
,
13000
0400
0040
00013
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,4],[13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[0,13,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,13] >;

C42.677C23 in GAP, Magma, Sage, TeX

C_4^2._{677}C_2^3
% in TeX

G:=Group("C4^2.677C2^3");
// GroupNames label

G:=SmallGroup(128,1652);
// by ID

G=gap.SmallGroup(128,1652);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,2019,124]);
// Polycyclic

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

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