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G = C42.304C23order 128 = 27

165th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.304C23, C4.1212- 1+4, (C8×Q8)⋊35C2, C4⋊Q8.36C4, C84Q842C2, C4.41(C8○D4), (C4×C8).31C22, C22⋊Q8.28C4, C4⋊C8.369C22, (C2×C4).683C24, (C2×C8).443C23, C42.231(C2×C4), C42.C2.21C4, (C4×Q8).284C22, C8⋊C4.102C22, C22⋊C8.239C22, C23.108(C22×C4), C22.206(C23×C4), (C22×C4).947C23, (C2×C42).790C22, C42.12C4.48C2, C42⋊C2.89C22, C42.7C22.4C2, C23.37C23.25C2, C2.26(C23.32C23), C2.34(C2×C8○D4), C4⋊C4.170(C2×C4), C22⋊C4.24(C2×C4), (C2×C4).85(C22×C4), (C2×Q8).167(C2×C4), (C22×C4).362(C2×C4), SmallGroup(128,1718)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.304C23
C1C2C4C2×C4C22×C4C2×C42C23.37C23 — C42.304C23
C1C22 — C42.304C23
C1C2×C4 — C42.304C23
C1C2C2C2×C4 — C42.304C23

Generators and relations for C42.304C23
 G = < a,b,c,d,e | a4=b4=1, c2=b, d2=e2=a2b2, ab=ba, ac=ca, dad-1=a-1, eae-1=ab2, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2b2c, ede-1=a2d >

Subgroups: 204 in 165 conjugacy classes, 128 normal (12 characteristic)
C1, C2, C2 [×2], C2, C4 [×6], C4 [×11], C22, C22 [×3], C8 [×8], C2×C4 [×2], C2×C4 [×12], C2×C4 [×6], Q8 [×8], C23, C42 [×2], C42 [×6], C22⋊C4 [×4], C4⋊C4 [×16], C2×C8 [×8], C22×C4, C22×C4 [×2], C2×Q8 [×4], C4×C8 [×8], C8⋊C4 [×4], C22⋊C8 [×4], C4⋊C8 [×12], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C42.12C4 [×2], C42.7C22 [×4], C8×Q8 [×4], C84Q8 [×4], C23.37C23, C42.304C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×4], C23×C4, 2- 1+4 [×2], C23.32C23, C2×C8○D4 [×2], C42.304C23

Smallest permutation representation of C42.304C23
On 64 points
Generators in S64
(1 59 51 47)(2 60 52 48)(3 61 53 41)(4 62 54 42)(5 63 55 43)(6 64 56 44)(7 57 49 45)(8 58 50 46)(9 18 34 30)(10 19 35 31)(11 20 36 32)(12 21 37 25)(13 22 38 26)(14 23 39 27)(15 24 40 28)(16 17 33 29)
(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)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 31 55 23)(2 32 56 24)(3 25 49 17)(4 26 50 18)(5 27 51 19)(6 28 52 20)(7 29 53 21)(8 30 54 22)(9 62 38 46)(10 63 39 47)(11 64 40 48)(12 57 33 41)(13 58 34 42)(14 59 35 43)(15 60 36 44)(16 61 37 45)
(1 63 55 47)(2 48 56 64)(3 57 49 41)(4 42 50 58)(5 59 51 43)(6 44 52 60)(7 61 53 45)(8 46 54 62)(9 26 38 18)(10 19 39 27)(11 28 40 20)(12 21 33 29)(13 30 34 22)(14 23 35 31)(15 32 36 24)(16 17 37 25)

G:=sub<Sym(64)| (1,59,51,47)(2,60,52,48)(3,61,53,41)(4,62,54,42)(5,63,55,43)(6,64,56,44)(7,57,49,45)(8,58,50,46)(9,18,34,30)(10,19,35,31)(11,20,36,32)(12,21,37,25)(13,22,38,26)(14,23,39,27)(15,24,40,28)(16,17,33,29), (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)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,31,55,23)(2,32,56,24)(3,25,49,17)(4,26,50,18)(5,27,51,19)(6,28,52,20)(7,29,53,21)(8,30,54,22)(9,62,38,46)(10,63,39,47)(11,64,40,48)(12,57,33,41)(13,58,34,42)(14,59,35,43)(15,60,36,44)(16,61,37,45), (1,63,55,47)(2,48,56,64)(3,57,49,41)(4,42,50,58)(5,59,51,43)(6,44,52,60)(7,61,53,45)(8,46,54,62)(9,26,38,18)(10,19,39,27)(11,28,40,20)(12,21,33,29)(13,30,34,22)(14,23,35,31)(15,32,36,24)(16,17,37,25)>;

G:=Group( (1,59,51,47)(2,60,52,48)(3,61,53,41)(4,62,54,42)(5,63,55,43)(6,64,56,44)(7,57,49,45)(8,58,50,46)(9,18,34,30)(10,19,35,31)(11,20,36,32)(12,21,37,25)(13,22,38,26)(14,23,39,27)(15,24,40,28)(16,17,33,29), (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)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,31,55,23)(2,32,56,24)(3,25,49,17)(4,26,50,18)(5,27,51,19)(6,28,52,20)(7,29,53,21)(8,30,54,22)(9,62,38,46)(10,63,39,47)(11,64,40,48)(12,57,33,41)(13,58,34,42)(14,59,35,43)(15,60,36,44)(16,61,37,45), (1,63,55,47)(2,48,56,64)(3,57,49,41)(4,42,50,58)(5,59,51,43)(6,44,52,60)(7,61,53,45)(8,46,54,62)(9,26,38,18)(10,19,39,27)(11,28,40,20)(12,21,33,29)(13,30,34,22)(14,23,35,31)(15,32,36,24)(16,17,37,25) );

G=PermutationGroup([(1,59,51,47),(2,60,52,48),(3,61,53,41),(4,62,54,42),(5,63,55,43),(6,64,56,44),(7,57,49,45),(8,58,50,46),(9,18,34,30),(10,19,35,31),(11,20,36,32),(12,21,37,25),(13,22,38,26),(14,23,39,27),(15,24,40,28),(16,17,33,29)], [(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),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,31,55,23),(2,32,56,24),(3,25,49,17),(4,26,50,18),(5,27,51,19),(6,28,52,20),(7,29,53,21),(8,30,54,22),(9,62,38,46),(10,63,39,47),(11,64,40,48),(12,57,33,41),(13,58,34,42),(14,59,35,43),(15,60,36,44),(16,61,37,45)], [(1,63,55,47),(2,48,56,64),(3,57,49,41),(4,42,50,58),(5,59,51,43),(6,44,52,60),(7,61,53,45),(8,46,54,62),(9,26,38,18),(10,19,39,27),(11,28,40,20),(12,21,33,29),(13,30,34,22),(14,23,35,31),(15,32,36,24),(16,17,37,25)])

50 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E···4L4M···4U8A···8P8Q···8X
order1222244444···44···48···88···8
size1111411112···24···42···24···4

50 irreducible representations

dim11111111124
type++++++-
imageC1C2C2C2C2C2C4C4C4C8○D42- 1+4
kernelC42.304C23C42.12C4C42.7C22C8×Q8C84Q8C23.37C23C22⋊Q8C42.C2C4⋊Q8C4C4
# reps124441844162

Matrix representation of C42.304C23 in GL4(𝔽17) generated by

161500
0100
001315
0004
,
4000
0400
00130
00013
,
2400
01500
0080
0008
,
4000
0400
00160
0041
,
161500
1100
001315
00164
G:=sub<GL(4,GF(17))| [16,0,0,0,15,1,0,0,0,0,13,0,0,0,15,4],[4,0,0,0,0,4,0,0,0,0,13,0,0,0,0,13],[2,0,0,0,4,15,0,0,0,0,8,0,0,0,0,8],[4,0,0,0,0,4,0,0,0,0,16,4,0,0,0,1],[16,1,0,0,15,1,0,0,0,0,13,16,0,0,15,4] >;

C42.304C23 in GAP, Magma, Sage, TeX

C_4^2._{304}C_2^3
% in TeX

G:=Group("C4^2.304C2^3");
// GroupNames label

G:=SmallGroup(128,1718);
// by ID

G=gap.SmallGroup(128,1718);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,891,100,675,1018,124]);
// Polycyclic

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

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