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G = C42.308C23order 128 = 27

169th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.308C23, C4.1812+ 1+4, C4.1282- 1+4, (C8×D4)⋊52C2, (C8×Q8)⋊37C2, C89D449C2, C86D447C2, C84Q847C2, C4.23(C8○D4), C4⋊D4.31C4, C22⋊Q8.31C4, C4⋊C8.241C22, (C4×C8).347C22, (C2×C4).690C24, (C2×C8).448C23, C422C2.5C4, C42.238(C2×C4), C4.4D4.24C4, (C4×D4).65C22, C42.C2.24C4, C4⋊M4(2)⋊39C2, C23.47(C22×C4), (C4×Q8).287C22, C42.12C459C2, C8⋊C4.107C22, C22⋊C8.150C22, C2.38(Q8○M4(2)), C22.212(C23×C4), (C22×C4).950C23, (C2×C42).797C22, (C22×C8).455C22, C22.D4.12C4, C42.6C2233C2, C42.7C2230C2, C42⋊C2.92C22, (C2×M4(2)).253C22, C23.36C23.17C2, C2.48(C23.33C23), C2.38(C2×C8○D4), C4⋊C4.171(C2×C4), (C2×D4).146(C2×C4), C22⋊C4.26(C2×C4), (C2×C4).88(C22×C4), (C2×Q8).168(C2×C4), (C22×C4).368(C2×C4), SmallGroup(128,1725)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.308C23
C1C2C4C2×C4C42C2×C42C23.36C23 — C42.308C23
C1C22 — C42.308C23
C1C2×C4 — C42.308C23
C1C2C2C2×C4 — C42.308C23

Generators and relations for C42.308C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, ac=ca, dad=a-1b2, eae=ab2, bc=cb, bd=db, be=eb, dcd=b2c, ece=a2b2c, ede=a2b2d >

Subgroups: 252 in 177 conjugacy classes, 126 normal (52 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×9], C8 [×8], C2×C4 [×6], C2×C4 [×6], C2×C4 [×9], D4 [×4], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], C2×C8 [×2], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4×C8 [×2], C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×2], C22⋊C8 [×4], C4⋊C8 [×4], C4⋊C8 [×6], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], C22×C8 [×2], C2×M4(2) [×4], C4⋊M4(2), C42.6C22 [×2], C42.12C4, C42.7C22 [×2], C8×D4, C89D4 [×2], C86D4, C86D4 [×2], C8×Q8, C84Q8, C23.36C23, C42.308C23
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- 1+4, C23.33C23, C2×C8○D4, Q8○M4(2), C42.308C23

Smallest permutation representation of C42.308C23
On 64 points
Generators in S64
(1 39 51 45)(2 40 52 46)(3 33 53 47)(4 34 54 48)(5 35 55 41)(6 36 56 42)(7 37 49 43)(8 38 50 44)(9 32 60 22)(10 25 61 23)(11 26 62 24)(12 27 63 17)(13 28 64 18)(14 29 57 19)(15 30 58 20)(16 31 59 21)
(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)
(2 6)(4 8)(9 60)(10 57)(11 62)(12 59)(13 64)(14 61)(15 58)(16 63)(18 22)(20 24)(26 30)(28 32)(33 43)(34 48)(35 45)(36 42)(37 47)(38 44)(39 41)(40 46)(50 54)(52 56)
(1 16)(2 64)(3 10)(4 58)(5 12)(6 60)(7 14)(8 62)(9 56)(11 50)(13 52)(15 54)(17 45)(18 36)(19 47)(20 38)(21 41)(22 40)(23 43)(24 34)(25 37)(26 48)(27 39)(28 42)(29 33)(30 44)(31 35)(32 46)(49 57)(51 59)(53 61)(55 63)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I···4Q8A···8H8I···8T
order1222222444444444···48···88···8
size1111444111122224···42···24···4

44 irreducible representations

dim111111111111111112444
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4C4C4C4C4C4C8○D42+ 1+42- 1+4Q8○M4(2)
kernelC42.308C23C4⋊M4(2)C42.6C22C42.12C4C42.7C22C8×D4C89D4C86D4C8×Q8C84Q8C23.36C23C4⋊D4C22⋊Q8C22.D4C4.4D4C42.C2C422C2C4C4C4C2
# reps112121231112242248112

Matrix representation of C42.308C23 in GL6(𝔽17)

020000
900000
000400
004000
001510413
0088013
,
400000
040000
0016000
0001600
0000160
0000016
,
040000
100000
00713150
00142152
0014161413
002161311
,
100000
0160000
001000
0001600
000410
0034216
,
080000
1500000
000100
001000
00513161
009801

G:=sub<GL(6,GF(17))| [0,9,0,0,0,0,2,0,0,0,0,0,0,0,0,4,15,8,0,0,4,0,10,8,0,0,0,0,4,0,0,0,0,0,13,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,7,14,14,2,0,0,13,2,16,16,0,0,15,15,14,13,0,0,0,2,13,11],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,0,16,4,4,0,0,0,0,1,2,0,0,0,0,0,16],[0,15,0,0,0,0,8,0,0,0,0,0,0,0,0,1,5,9,0,0,1,0,13,8,0,0,0,0,16,0,0,0,0,0,1,1] >;

C42.308C23 in GAP, Magma, Sage, TeX

C_4^2._{308}C_2^3
% in TeX

G:=Group("C4^2.308C2^3");
// GroupNames label

G:=SmallGroup(128,1725);
// by ID

G=gap.SmallGroup(128,1725);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,219,675,1018,80,124]);
// Polycyclic

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

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