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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, Q165(C2×C4), C4.78(C4×D4), C4⋊C4.399D4, (C4×Q16)⋊28C2, SD164(C2×C4), C8.C225C4, C8.3(C22×C4), C2.3(Q8○D8), Q16⋊C42C2, (C4×SD16)⋊14C2, C4.24(C23×C4), D4.7(C22×C4), C22.17(C4×D4), SD16⋊C42C2, Q8.7(C22×C4), C4⋊C4.364C23, C82M4(2)⋊5C2, M4(2)⋊10(C2×C4), (C2×C4).204C24, (C2×C8).415C23, (C4×C8).174C22, C22⋊C4.186D4, C2.5(D4○SD16), (C4×D4).57C22, C23.436(C2×D4), (C4×Q8).53C22, (C2×D4).373C23, (C2×Q8).346C23, M4(2)⋊C411C2, C2.D8.213C22, C8⋊C4.113C22, C4.Q8.127C22, (C22×C4).925C23, (C22×C8).441C22, (C2×Q16).153C22, C22.148(C22×D4), D4⋊C4.216C22, C23.32C235C2, C42⋊C2.81C22, Q8⋊C4.197C22, (C2×SD16).110C22, C23.24D4.11C2, (C22×Q8).258C22, (C2×M4(2)).261C22, C23.33C23.6C2, C2.64(C2×C4×D4), (C2×Q8)⋊20(C2×C4), C4.12(C2×C4○D4), C4○D4.10(C2×C4), (C2×C4).911(C2×D4), (C2×Q8⋊C4)⋊53C2, (C2×C4).71(C22×C4), (C2×C8.C22).9C2, (C2×C4).475(C4○D4), (C2×C4⋊C4).575C22, (C2×C4○D4).88C22, SmallGroup(128,1679)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42.276C23
C1C2C22C2×C4C22×C4C42⋊C2C23.32C23 — C42.276C23
C1C2C4 — C42.276C23
C1C22C42⋊C2 — C42.276C23
C1C2C2C2×C4 — 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 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×14], C22, C22 [×2], C22 [×6], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×18], D4 [×2], D4 [×5], Q8 [×6], Q8 [×7], C23, C23, C42 [×2], C42 [×7], C22⋊C4 [×2], C22⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×4], C4○D4 [×2], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2 [×2], C42⋊C2 [×3], C4×D4 [×2], C4×D4 [×2], C4×Q8 [×6], C4×Q8 [×2], C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C82M4(2), C2×Q8⋊C4, C23.24D4, M4(2)⋊C4, C4×SD16 [×2], C4×Q16 [×2], SD16⋊C4 [×2], Q16⋊C4 [×2], C23.32C23, C23.33C23, C2×C8.C22, C42.276C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, D4○SD16, Q8○D8, 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++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C4D4D4C4○D4D4○SD16Q8○D8
kernelC42.276C23C82M4(2)C2×Q8⋊C4C23.24D4M4(2)⋊C4C4×SD16C4×Q16SD16⋊C4Q16⋊C4C23.32C23C23.33C23C2×C8.C22C8.C22C22⋊C4C4⋊C4C2×C4C2C2
# 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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