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G = C42.309C23order 128 = 27

170th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.309C23, C4.1292- 1+4, C4.1822+ 1+4, (C8×D4)⋊53C2, (C8×Q8)⋊38C2, C86D448C2, C89D450C2, C84Q848C2, C4.42(C8○D4), C4⋊D4.32C4, C22⋊Q8.32C4, C4⋊C8.373C22, (C2×C4).691C24, C42.239(C2×C4), C422C2.6C4, (C2×C8).449C23, (C4×C8).348C22, C4.4D4.25C4, C22.8(C8○D4), C42.C2.25C4, (C4×D4).307C22, C23.48(C22×C4), (C22×C8).96C22, (C4×Q8).288C22, C8⋊C4.108C22, C42.12C460C2, C22⋊C8.242C22, (C22×C4).951C23, C22.213(C23×C4), (C2×C42).798C22, C22.D4.13C4, C42⋊C2.93C22, C42.7C2231C2, C42.6C2234C2, (C2×M4(2)).254C22, C23.36C23.18C2, C2.49(C23.33C23), (C2×C4⋊C8)⋊53C2, C2.39(C2×C8○D4), C4⋊C4.172(C2×C4), (C2×D4).186(C2×C4), C22⋊C4.46(C2×C4), (C2×C4).89(C22×C4), (C2×Q8).169(C2×C4), (C22×C4).369(C2×C4), SmallGroup(128,1726)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.309C23
C1C2C4C2×C4C42C2×C42C23.36C23 — C42.309C23
C1C22 — C42.309C23
C1C2×C4 — C42.309C23
C1C2C2C2×C4 — C42.309C23

Generators and relations for C42.309C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, ac=ca, dad=a-1, eae=ab2, bc=cb, bd=db, be=eb, cd=dc, ece=a2b2c, ede=a2d >

Subgroups: 252 in 182 conjugacy classes, 128 normal (52 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×6], C2×C4 [×6], C2×C4 [×10], D4 [×6], 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 [×4], M4(2) [×2], 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 [×4], C2×M4(2) [×2], C2×C4⋊C8, C42.6C22 [×2], C42.12C4, C42.7C22 [×2], C8×D4, C8×D4 [×2], C89D4 [×2], C86D4, C8×Q8, C84Q8, C23.36C23, C42.309C23
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- 1+4, C23.33C23, C2×C8○D4 [×2], C42.309C23

Smallest permutation representation of C42.309C23
On 64 points
Generators in S64
(1 41 51 59)(2 42 52 60)(3 43 53 61)(4 44 54 62)(5 45 55 63)(6 46 56 64)(7 47 49 57)(8 48 50 58)(9 21 32 40)(10 22 25 33)(11 23 26 34)(12 24 27 35)(13 17 28 36)(14 18 29 37)(15 19 30 38)(16 20 31 39)
(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 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(1 20)(2 36)(3 22)(4 38)(5 24)(6 40)(7 18)(8 34)(9 42)(10 57)(11 44)(12 59)(13 46)(14 61)(15 48)(16 63)(17 52)(19 54)(21 56)(23 50)(25 47)(26 62)(27 41)(28 64)(29 43)(30 58)(31 45)(32 60)(33 53)(35 55)(37 49)(39 51)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4J4K···4R8A···8P8Q···8X
order1222222244444···44···48···88···8
size1111224411112···24···42···24···4

50 irreducible representations

dim111111111111111112244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4C4C4C4C4C4C8○D4C8○D42+ 1+42- 1+4
kernelC42.309C23C2×C4⋊C8C42.6C22C42.12C4C42.7C22C8×D4C89D4C86D4C8×Q8C84Q8C23.36C23C4⋊D4C22⋊Q8C22.D4C4.4D4C42.C2C422C2C4C22C4C4
# reps112123211112242248811

Matrix representation of C42.309C23 in GL4(𝔽17) generated by

11500
11600
00160
0001
,
4000
0400
00130
00013
,
15000
01500
0080
0009
,
11500
01600
00160
00016
,
16000
16100
00016
00160
G:=sub<GL(4,GF(17))| [1,1,0,0,15,16,0,0,0,0,16,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,13,0,0,0,0,13],[15,0,0,0,0,15,0,0,0,0,8,0,0,0,0,9],[1,0,0,0,15,16,0,0,0,0,16,0,0,0,0,16],[16,16,0,0,0,1,0,0,0,0,0,16,0,0,16,0] >;

C42.309C23 in GAP, Magma, Sage, TeX

C_4^2._{309}C_2^3
% in TeX

G:=Group("C4^2.309C2^3");
// GroupNames label

G:=SmallGroup(128,1726);
// by ID

G=gap.SmallGroup(128,1726);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,891,675,1018,80,124]);
// 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,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^2*b^2*c,e*d*e=a^2*d>;
// generators/relations

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