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G = C42.297C23order 128 = 27

158th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.297C23, C4.1702+ 1+4, (C8×D4)⋊46C2, C86D441C2, C89D441C2, C4⋊C852C22, (C4×C8)⋊61C22, C22≀C2.5C4, C4⋊D4.24C4, C24.87(C2×C4), C8⋊C431C22, C22⋊Q8.24C4, C22⋊C880C22, (C2×C8).434C23, (C2×C4).673C24, (C22×C8)⋊56C22, C22.4(C8○D4), C24.4C437C2, (C4×D4).300C22, C22.D4.8C4, C2.29(Q8○M4(2)), (C2×M4(2))⋊47C22, (C23×C4).532C22, C23.150(C22×C4), C22.197(C23×C4), C42⋊C2.85C22, (C22×C4).1282C23, C42.7C2226C2, C22.19C24.12C2, C2.47(C22.11C24), C2.28(C2×C8○D4), C4⋊C4.167(C2×C4), (C2×C22⋊C8)⋊47C2, (C2×D4).182(C2×C4), C22⋊C4.19(C2×C4), (C2×C4).79(C22×C4), (C2×Q8).122(C2×C4), (C22×C8)⋊C232C2, (C22×C4).354(C2×C4), (C2×C4○D4).93C22, SmallGroup(128,1708)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.297C23
C1C2C4C2×C4C22×C4C23×C4C22.19C24 — C42.297C23
C1C22 — C42.297C23
C1C2×C4 — C42.297C23
C1C2C2C2×C4 — C42.297C23

Generators and relations for C42.297C23
 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=a2c, ede=b2d >

Subgroups: 348 in 206 conjugacy classes, 126 normal (40 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×18], C8 [×8], C2×C4 [×4], C2×C4 [×6], C2×C4 [×13], D4 [×9], Q8, C23 [×3], C23 [×2], C23 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×6], C22×C4 [×2], 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], C22×C8 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C23×C4, C2×C4○D4, C2×C22⋊C8, C24.4C4, (C22×C8)⋊C2 [×2], C42.7C22 [×2], C8×D4 [×2], C89D4 [×4], C86D4 [×2], C22.19C24, C42.297C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×2], C23×C4, 2+ 1+4 [×2], C22.11C24, C2×C8○D4, Q8○M4(2), C42.297C23

Smallest permutation representation of C42.297C23
On 32 points
Generators in S32
(1 19 31 10)(2 11 32 20)(3 21 25 12)(4 13 26 22)(5 23 27 14)(6 15 28 24)(7 17 29 16)(8 9 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)
(1 10)(2 20)(3 12)(4 22)(5 14)(6 24)(7 16)(8 18)(9 30)(11 32)(13 26)(15 28)(17 29)(19 31)(21 25)(23 27)
(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(9 18)(11 20)(13 22)(15 24)(25 29)(27 31)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G···4N8A···8H8I···8T
order12222222224444444···48···88···8
size11112244441111224···42···24···4

44 irreducible representations

dim1111111111111244
type++++++++++
imageC1C2C2C2C2C2C2C2C2C4C4C4C4C8○D42+ 1+4Q8○M4(2)
kernelC42.297C23C2×C22⋊C8C24.4C4(C22×C8)⋊C2C42.7C22C8×D4C89D4C86D4C22.19C24C22≀C2C4⋊D4C22⋊Q8C22.D4C22C4C2
# reps1112224214444822

Matrix representation of C42.297C23 in GL6(𝔽17)

16150000
010000
000100
0016000
001111
000151516
,
1300000
0130000
001000
000100
000010
000001
,
800000
080000
000010
0016161616
001000
000001
,
16150000
010000
000100
001000
001111
001515016
,
1600000
110000
001000
000100
0000160
001515016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,16,1,0,0,0,1,0,1,15,0,0,0,0,1,15,0,0,0,0,1,16],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,16,1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,0,0,16,0,1],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,1,1,15,0,0,1,0,1,15,0,0,0,0,1,0,0,0,0,0,1,16],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,15,0,0,0,1,0,15,0,0,0,0,16,0,0,0,0,0,0,16] >;

C42.297C23 in GAP, Magma, Sage, TeX

C_4^2._{297}C_2^3
% in TeX

G:=Group("C4^2.297C2^3");
// GroupNames label

G:=SmallGroup(128,1708);
// by ID

G=gap.SmallGroup(128,1708);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,1018,124]);
// 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*c,e*d*e=b^2*d>;
// generators/relations

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