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G = C24.85D4order 128 = 27

40th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.85D4, C23.18SD16, C4⋊C4.90D4, (C2×C8).156D4, (C2×Q8).90D4, C2.18(C88D4), (C22×C4).147D4, C23.911(C2×D4), C4.144(C4⋊D4), C22.4Q1623C2, C2.21(Q8⋊D4), C2.13(C8.D4), C4.38(C4.4D4), (C22×C8).71C22, C22.96(C2×SD16), C22.217C22≀C2, C2.32(D4.7D4), C22.109(C4○D8), (C23×C4).273C22, C23.7Q8.18C2, (C22×Q8).63C22, C22.228(C4⋊D4), (C22×C4).1445C23, C2.8(C23.20D4), C2.6(C23.47D4), C4.19(C22.D4), C2.8(C23.10D4), C22.125(C8.C22), C22.114(C22.D4), (C2×C4.Q8)⋊21C2, (C2×Q8⋊C4)⋊13C2, (C2×C4).1037(C2×D4), (C2×C22⋊C8).37C2, (C2×C22⋊Q8).13C2, (C2×C4).772(C4○D4), (C2×C4⋊C4).120C22, SmallGroup(128,767)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C24.85D4
C1C2C22C2×C4C22×C4C2×C4⋊C4C23.7Q8 — C24.85D4
C1C2C22×C4 — C24.85D4
C1C23C23×C4 — C24.85D4
C1C2C2C22×C4 — C24.85D4

Generators and relations for C24.85D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, faf-1=ab=ba, ac=ca, ad=da, eae-1=abc, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be3 >

Subgroups: 336 in 158 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×7], C22 [×7], C22 [×10], C8 [×3], C2×C4 [×6], C2×C4 [×21], Q8 [×6], C23, C23 [×2], C23 [×6], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×2], C2×C8 [×5], C22×C4 [×2], C22×C4 [×10], C2×Q8 [×2], C2×Q8 [×5], C24, C2.C42, C22⋊C8 [×2], Q8⋊C4 [×4], C4.Q8 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4, C22⋊Q8 [×4], C22×C8 [×2], C23×C4, C22×Q8, C22.4Q16, C23.7Q8, C2×C22⋊C8, C2×Q8⋊C4 [×2], C2×C4.Q8, C2×C22⋊Q8, C24.85D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, SD16 [×2], C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C2×SD16, C4○D8, C8.C22 [×2], C23.10D4, Q8⋊D4, D4.7D4, C88D4, C8.D4, C23.47D4, C23.20D4, C24.85D4

Smallest permutation representation of C24.85D4
On 64 points
Generators in S64
(2 14)(4 16)(6 10)(8 12)(17 53)(18 38)(19 55)(20 40)(21 49)(22 34)(23 51)(24 36)(26 64)(28 58)(30 60)(32 62)(33 47)(35 41)(37 43)(39 45)(42 52)(44 54)(46 56)(48 50)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 49)(48 50)
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)(17 53)(18 54)(19 55)(20 56)(21 49)(22 50)(23 51)(24 52)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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 20 5 24)(2 35 6 39)(3 18 7 22)(4 33 8 37)(9 42 13 46)(10 55 14 51)(11 48 15 44)(12 53 16 49)(17 26 21 30)(19 32 23 28)(25 38 29 34)(27 36 31 40)(41 58 45 62)(43 64 47 60)(50 63 54 59)(52 61 56 57)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N8A···8H
order12···2224444444···48···8
size11···1442222448···84···4

32 irreducible representations

dim1111111222222224
type++++++++++++-
imageC1C2C2C2C2C2C2D4D4D4D4D4C4○D4SD16C4○D8C8.C22
kernelC24.85D4C22.4Q16C23.7Q8C2×C22⋊C8C2×Q8⋊C4C2×C4.Q8C2×C22⋊Q8C4⋊C4C2×C8C22×C4C2×Q8C24C2×C4C23C22C22
# reps1111211221216442

Matrix representation of C24.85D4 in GL6(𝔽17)

100000
0160000
001000
0001600
000010
0000316
,
1600000
0160000
0016000
0001600
000010
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
000010
000001
,
1300000
0130000
002000
000900
0000315
0000514
,
040000
1300000
000800
002000
000010
0000316

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

C24.85D4 in GAP, Magma, Sage, TeX

C_2^4._{85}D_4
% in TeX

G:=Group("C2^4.85D4");
// GroupNames label

G:=SmallGroup(128,767);
// by ID

G=gap.SmallGroup(128,767);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,58,2804,718,172]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=f^2=d,f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*e^3>;
// generators/relations

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