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G = C42.298C23order 128 = 27

159th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.298C23, C4.1712+ 1+4, (C8×D4)⋊47C2, (C4×C8)⋊5C22, C89D442C2, C4⋊C892C22, C22≀C2.6C4, C4⋊D4.25C4, C24.88(C2×C4), (C22×C8)⋊6C22, C8⋊C432C22, C22⋊Q8.25C4, C22⋊C881C22, (C2×C4).674C24, (C2×C8).435C23, C22.7(C8○D4), (C4×D4).301C22, C23.41(C22×C4), C22.D4.9C4, (C2×M4(2))⋊48C22, (C22×C4).941C23, C22.198(C23×C4), (C23×C4).533C22, C42⋊C2.86C22, C42.7C2227C2, C22.19C24.13C2, C2.48(C22.11C24), C2.29(C2×C8○D4), C4⋊C4.168(C2×C4), (C2×C22⋊C8)⋊48C2, (C2×D4).183(C2×C4), C22⋊C4.43(C2×C4), (C2×C4).80(C22×C4), (C2×Q8).123(C2×C4), (C22×C8)⋊C233C2, (C22×C4).139(C2×C4), (C2×C4○D4).94C22, SmallGroup(128,1709)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.298C23
C1C2C4C2×C4C22×C4C23×C4C22.19C24 — C42.298C23
C1C22 — C42.298C23
C1C2×C4 — C42.298C23
C1C2C2C2×C4 — C42.298C23

Generators and relations for C42.298C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, dad=a-1, eae=ab2, bc=cb, bd=db, be=eb, dcd=ece=a2b2c, ede=b2d >

Subgroups: 348 in 211 conjugacy classes, 128 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22×C8, C2×M4(2), C23×C4, C2×C4○D4, C2×C22⋊C8, (C22×C8)⋊C2, C42.7C22, C8×D4, C89D4, C22.19C24, C42.298C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C8○D4, C23×C4, 2+ 1+4, C22.11C24, C2×C8○D4, C42.298C23

Smallest permutation representation of C42.298C23
On 32 points
Generators in S32
(1 23 27 15)(2 12 28 20)(3 17 29 9)(4 14 30 22)(5 19 31 11)(6 16 32 24)(7 21 25 13)(8 10 26 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)
(1 15)(2 20)(3 9)(4 22)(5 11)(6 24)(7 13)(8 18)(10 26)(12 28)(14 30)(16 32)(17 29)(19 31)(21 25)(23 27)
(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(10 22)(12 24)(14 18)(16 20)(25 29)(27 31)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I···4O8A···8P8Q···8X
order12222222222444444444···48···88···8
size11112222444111122224···42···24···4

50 irreducible representations

dim1111111111124
type++++++++
imageC1C2C2C2C2C2C2C4C4C4C4C8○D42+ 1+4
kernelC42.298C23C2×C22⋊C8(C22×C8)⋊C2C42.7C22C8×D4C89D4C22.19C24C22≀C2C4⋊D4C22⋊Q8C22.D4C22C4
# reps12224414444162

Matrix representation of C42.298C23 in GL4(𝔽17) generated by

11500
11600
0001
0010
,
4000
0400
0040
0004
,
2000
0200
0008
0090
,
11500
01600
0001
0010
,
16000
16100
00160
0001
G:=sub<GL(4,GF(17))| [1,1,0,0,15,16,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,2,0,0,0,0,0,9,0,0,8,0],[1,0,0,0,15,16,0,0,0,0,0,1,0,0,1,0],[16,16,0,0,0,1,0,0,0,0,16,0,0,0,0,1] >;

C42.298C23 in GAP, Magma, Sage, TeX

C_4^2._{298}C_2^3
% in TeX

G:=Group("C4^2.298C2^3");
// GroupNames label

G:=SmallGroup(128,1709);
// by ID

G=gap.SmallGroup(128,1709);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,891,675,1018,124]);
// 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*b^2,d*a*d=a^-1,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=e*c*e=a^2*b^2*c,e*d*e=b^2*d>;
// generators/relations

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