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G = C42.276C23order 128 = 27

137th non-split extension by C42 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.276C23, Q16:5(C2xC4), C4.78(C4xD4), C4:C4.399D4, (C4xQ16):28C2, SD16:4(C2xC4), C8.C22:5C4, C8.3(C22xC4), C2.3(Q8oD8), Q16:C4:2C2, (C4xSD16):14C2, C4.24(C23xC4), D4.7(C22xC4), C22.17(C4xD4), SD16:C4:2C2, Q8.7(C22xC4), C4:C4.364C23, C8o2M4(2):5C2, M4(2):10(C2xC4), (C2xC4).204C24, (C2xC8).415C23, (C4xC8).174C22, C22:C4.186D4, C2.5(D4oSD16), (C4xD4).57C22, C23.436(C2xD4), (C4xQ8).53C22, (C2xD4).373C23, (C2xQ8).346C23, M4(2):C4:11C2, C2.D8.213C22, C8:C4.113C22, C4.Q8.127C22, (C22xC4).925C23, (C22xC8).441C22, (C2xQ16).153C22, C22.148(C22xD4), D4:C4.216C22, C23.32C23:5C2, C42:C2.81C22, Q8:C4.197C22, (C2xSD16).110C22, C23.24D4.11C2, (C22xQ8).258C22, (C2xM4(2)).261C22, C23.33C23.6C2, C2.64(C2xC4xD4), (C2xQ8):20(C2xC4), C4.12(C2xC4oD4), C4oD4.10(C2xC4), (C2xC4).911(C2xD4), (C2xQ8:C4):53C2, (C2xC4).71(C22xC4), (C2xC8.C22).9C2, (C2xC4).475(C4oD4), (C2xC4:C4).575C22, (C2xC4oD4).88C22, SmallGroup(128,1679)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42.276C23
C1C2C22C2xC4C22xC4C42:C2C23.32C23 — C42.276C23
C1C2C4 — C42.276C23
C1C22C42:C2 — C42.276C23
C1C2C2C2xC4 — C42.276C23

Generators and relations for C42.276C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=b2, ab=ba, ac=ca, ad=da, eae=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ce=ec, de=ed >

Subgroups: 364 in 235 conjugacy classes, 140 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, C2xC8, M4(2), SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C4oD4, C4xC8, C8:C4, D4:C4, Q8:C4, Q8:C4, C4.Q8, C2.D8, C2xC4:C4, C2xC4:C4, C42:C2, C42:C2, C4xD4, C4xD4, C4xQ8, C4xQ8, C22xC8, C2xM4(2), C2xSD16, C2xQ16, C8.C22, C22xQ8, C2xC4oD4, C8o2M4(2), C2xQ8:C4, C23.24D4, M4(2):C4, C4xSD16, C4xQ16, SD16:C4, Q16:C4, C23.32C23, C23.33C23, C2xC8.C22, C42.276C23
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, C24, C4xD4, C23xC4, C22xD4, C2xC4oD4, C2xC4xD4, D4oSD16, Q8oD8, C42.276C23

Smallest permutation representation of C42.276C23
On 64 points
Generators in S64
(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 15 27 23)(2 16 28 24)(3 13 25 21)(4 14 26 22)(5 11 35 31)(6 12 36 32)(7 9 33 29)(8 10 34 30)(17 42 49 62)(18 43 50 63)(19 44 51 64)(20 41 52 61)(37 53 57 45)(38 54 58 46)(39 55 59 47)(40 56 60 48)
(1 4 3 2)(5 8 7 6)(9 32 11 30)(10 29 12 31)(13 24 15 22)(14 21 16 23)(17 41 19 43)(18 42 20 44)(25 28 27 26)(33 36 35 34)(37 56 39 54)(38 53 40 55)(45 60 47 58)(46 57 48 59)(49 61 51 63)(50 62 52 64)
(1 59 27 39)(2 60 28 40)(3 57 25 37)(4 58 26 38)(5 18 35 50)(6 19 36 51)(7 20 33 52)(8 17 34 49)(9 61 29 41)(10 62 30 42)(11 63 31 43)(12 64 32 44)(13 53 21 45)(14 54 22 46)(15 55 23 47)(16 56 24 48)
(1 11)(2 32)(3 9)(4 30)(5 23)(6 16)(7 21)(8 14)(10 26)(12 28)(13 33)(15 35)(17 54)(18 47)(19 56)(20 45)(22 34)(24 36)(25 29)(27 31)(37 41)(38 62)(39 43)(40 64)(42 58)(44 60)(46 49)(48 51)(50 55)(52 53)(57 61)(59 63)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4L4M···4Z8A8B8C8D8E···8J
order122222224···44···488888···8
size111122442···24···422224···4

44 irreducible representations

dim111111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C4D4D4C4oD4D4oSD16Q8oD8
kernelC42.276C23C8o2M4(2)C2xQ8:C4C23.24D4M4(2):C4C4xSD16C4xQ16SD16:C4Q16:C4C23.32C23C23.33C23C2xC8.C22C8.C22C22:C4C4:C4C2xC4C2C2
# reps1111122221111622422

Matrix representation of C42.276C23 in GL6(F17)

1300000
0130000
0070016
000710
0001100
00160010
,
1600000
0160000
0001600
001000
0000016
000010
,
400000
12130000
00010160
0010001
00160010
0001100
,
6130000
13110000
0044314
004131414
0031444
001414413
,
1600000
0160000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,7,0,0,16,0,0,0,7,1,0,0,0,0,1,10,0,0,0,16,0,0,10],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[4,12,0,0,0,0,0,13,0,0,0,0,0,0,0,10,16,0,0,0,10,0,0,1,0,0,16,0,0,10,0,0,0,1,10,0],[6,13,0,0,0,0,13,11,0,0,0,0,0,0,4,4,3,14,0,0,4,13,14,14,0,0,3,14,4,4,0,0,14,14,4,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C42.276C23 in GAP, Magma, Sage, TeX

C_4^2._{276}C_2^3
% in TeX

G:=Group("C4^2.276C2^3");
// GroupNames label

G:=SmallGroup(128,1679);
// by ID

G=gap.SmallGroup(128,1679);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,184,521,2804,1411,172]);
// Polycyclic

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

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