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G = C42.299C23order 128 = 27

160th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.299C23, C4.1722+ 1+4, C89D443C2, C86D442C2, C4⋊C853C22, (C4×C8)⋊62C22, C22≀C2.7C4, C4⋊D4.26C4, C24.89(C2×C4), C8⋊C433C22, C22⋊Q8.26C4, C22⋊C847C22, (C2×C8).436C23, (C2×C4).675C24, (C22×C8)⋊57C22, (C4×D4).63C22, C24.4C438C2, C23.42(C22×C4), C2.30(Q8○M4(2)), (C2×M4(2))⋊49C22, C22.199(C23×C4), (C23×C4).534C22, (C22×C4).942C23, C22.D4.10C4, C42⋊C2.87C22, C42.7C2228C2, C22.19C24.14C2, C2.49(C22.11C24), C4⋊C4.119(C2×C4), (C2×D4).143(C2×C4), C22⋊C4.20(C2×C4), (C2×C4).81(C22×C4), (C2×Q8).124(C2×C4), (C22×C8)⋊C234C2, (C22×C4).355(C2×C4), (C2×C4○D4).95C22, SmallGroup(128,1710)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.299C23
C1C2C4C2×C4C22×C4C23×C4C22.19C24 — C42.299C23
C1C22 — C42.299C23
C1C2×C4 — C42.299C23
C1C2C2C2×C4 — C42.299C23

Generators and relations for C42.299C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=dad=a-1, eae=ab2, bc=cb, bd=db, be=eb, dcd=ece=a2b2c, ede=b2d >

Subgroups: 348 in 201 conjugacy classes, 124 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×19], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×7], Q8, C23, C23 [×4], C23 [×3], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×2], M4(2) [×6], C22×C4 [×2], C22×C4 [×6], C22×C4 [×2], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×2], C24, C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×12], C4⋊C8 [×4], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C22×C8 [×2], C2×M4(2) [×6], C23×C4, C2×C4○D4, C24.4C4 [×2], (C22×C8)⋊C2 [×2], C42.7C22 [×2], C89D4 [×4], C86D4 [×4], C22.19C24, C42.299C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ 1+4 [×2], C22.11C24, Q8○M4(2) [×2], C42.299C23

Smallest permutation representation of C42.299C23
On 32 points
Generators in S32
(1 23 27 14)(2 15 28 24)(3 17 29 16)(4 9 30 18)(5 19 31 10)(6 11 32 20)(7 21 25 12)(8 13 26 22)
(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 14)(2 20)(3 16)(4 22)(5 10)(6 24)(7 12)(8 18)(9 26)(11 28)(13 30)(15 32)(17 29)(19 31)(21 25)(23 27)
(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(9 22)(11 24)(13 18)(15 20)(25 29)(27 31)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D···2H4A4B4C4D4E···4M8A···8P
order12222···244444···48···8
size11114···411114···44···4

38 irreducible representations

dim1111111111144
type++++++++
imageC1C2C2C2C2C2C2C4C4C4C42+ 1+4Q8○M4(2)
kernelC42.299C23C24.4C4(C22×C8)⋊C2C42.7C22C89D4C86D4C22.19C24C22≀C2C4⋊D4C22⋊Q8C22.D4C4C2
# reps1222441444424

Matrix representation of C42.299C23 in GL8(𝔽17)

115000000
016000000
016010000
016100000
00000100
000016000
000000016
00000010
,
40000000
04000000
00400000
00040000
000013000
000001300
000000130
000000013
,
101500000
001610000
701600000
741600000
00000010
00000001
000013000
000001300
,
115000000
016000000
1160160000
1161600000
00000100
00001000
00000001
00000010
,
160000000
161000000
160100000
000160000
000016000
00000100
000000160
00000001

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[1,0,7,7,0,0,0,0,0,0,0,4,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,1,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[16,16,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1] >;

C42.299C23 in GAP, Magma, Sage, TeX

C_4^2._{299}C_2^3
% in TeX

G:=Group("C4^2.299C2^3");
// GroupNames label

G:=SmallGroup(128,1710);
// by ID

G=gap.SmallGroup(128,1710);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,891,675,1018,124]);
// 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=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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