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G = C42.305C23order 128 = 27

166th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.305C23, C4.1222- 1+4, C4⋊Q8.37C4, C84Q843C2, C22⋊Q8.29C4, C4⋊C8.239C22, (C2×C4).684C24, C42.232(C2×C4), (C4×C8).342C22, (C2×C8).444C23, C42.C2.22C4, (C4×Q8).61C22, C8⋊C4.103C22, C22⋊C8.147C22, C2.35(Q8○M4(2)), C42.6C4.35C2, (C2×C42).791C22, C22.207(C23×C4), (C22×C4).948C23, C23.109(C22×C4), C42⋊C2.90C22, C42.7C22.5C2, C23.37C23.26C2, C2.27(C23.32C23), C4⋊C4.123(C2×C4), C22⋊C4.25(C2×C4), (C2×C4).86(C22×C4), (C2×Q8).128(C2×C4), (C22×C4).363(C2×C4), SmallGroup(128,1719)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.305C23
C1C2C4C2×C4C22×C4C2×C42C23.37C23 — C42.305C23
C1C22 — C42.305C23
C1C2×C4 — C42.305C23
C1C2C2C2×C4 — C42.305C23

Generators and relations for C42.305C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=b, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, eae=ab2, bc=cb, bd=db, be=eb, dcd-1=b2c, ece=a2c, de=ed >

Subgroups: 204 in 157 conjugacy classes, 124 normal (12 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C8 [×8], C2×C4 [×2], C2×C4 [×12], C2×C4 [×4], Q8 [×4], C23, C42 [×2], C42 [×6], C22⋊C4 [×4], C4⋊C4 [×16], C2×C8 [×8], C22×C4, C22×C4 [×2], C2×Q8 [×4], C4×C8 [×4], C8⋊C4 [×8], C22⋊C8 [×4], C4⋊C8 [×12], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C42.6C4 [×2], C42.7C22 [×4], C84Q8 [×8], C23.37C23, C42.305C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2- 1+4 [×2], C23.32C23, Q8○M4(2) [×2], C42.305C23

Smallest permutation representation of C42.305C23
On 64 points
Generators in S64
(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)
(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 28 56 20)(3 25 49 17)(4 30 50 22)(5 27 51 19)(6 32 52 24)(7 29 53 21)(8 26 54 18)(9 58 38 42)(10 63 39 47)(11 60 40 44)(12 57 33 41)(13 62 34 46)(14 59 35 43)(15 64 36 48)(16 61 37 45)
(2 56)(4 50)(6 52)(8 54)(9 34)(10 14)(11 36)(12 16)(13 38)(15 40)(18 26)(20 28)(22 30)(24 32)(33 37)(35 39)(41 45)(42 62)(43 47)(44 64)(46 58)(48 60)(57 61)(59 63)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E···4Q8A···8P
order1222244444···48···8
size1111411114···44···4

38 irreducible representations

dim1111111144
type+++++-
imageC1C2C2C2C2C4C4C42- 1+4Q8○M4(2)
kernelC42.305C23C42.6C4C42.7C22C84Q8C23.37C23C22⋊Q8C42.C2C4⋊Q8C4C2
# reps1248184424

Matrix representation of C42.305C23 in GL8(𝔽17)

101500000
001610000
101600000
1161600000
000010150
000000131
000000160
000001130
,
130000000
013000000
001300000
000130000
00004000
00000400
00000040
00000004
,
901600000
90880000
00800000
98800000
00004400
0000141300
0000021215
000014825
,
115000000
116000000
016010000
1161600000
000041500
0000161300
0000416016
0000013160
,
10000000
01000000
101600000
100160000
00001000
00000100
000010160
000040016

G:=sub<GL(8,GF(17))| [1,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,13,16,13,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[9,9,0,9,0,0,0,0,0,0,0,8,0,0,0,0,16,8,8,8,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,4,14,0,14,0,0,0,0,4,13,2,8,0,0,0,0,0,0,12,2,0,0,0,0,0,0,15,5],[1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,16,4,0,0,0,0,0,15,13,16,13,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,1,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;

C42.305C23 in GAP, Magma, Sage, TeX

C_4^2._{305}C_2^3
% in TeX

G:=Group("C4^2.305C2^3");
// GroupNames label

G:=SmallGroup(128,1719);
// by ID

G=gap.SmallGroup(128,1719);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,1430,891,100,675,1018,124]);
// Polycyclic

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

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