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G = C24.88D4order 128 = 27

43rd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.88D4, C23.9Q16, (C2×C8).54D4, C2.16(C82D4), (C22×C4).161D4, C23.931(C2×D4), C22.61(C2×Q16), C22.4Q1630C2, C4.55(C4.4D4), C4.19(C422C2), C2.16(C8.18D4), C22.123(C4○D8), (C23×C4).276C22, (C22×C8).116C22, C23.7Q8.21C2, C22.252(C4⋊D4), C22.151(C8⋊C22), (C22×C4).1465C23, C2.8(C23.48D4), C2.11(C23.11D4), C2.11(C23.19D4), C4.111(C22.D4), C22.121(C22.D4), (C2×C2.D8)⋊11C2, (C2×C4).1374(C2×D4), (C2×C22⋊C8).29C2, (C2×C4).627(C4○D4), (C2×C4⋊C4).150C22, SmallGroup(128,808)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C24.88D4
C1C2C22C2×C4C22×C4C2×C4⋊C4C23.7Q8 — C24.88D4
C1C2C22×C4 — C24.88D4
C1C23C23×C4 — C24.88D4
C1C2C2C22×C4 — C24.88D4

Generators and relations for C24.88D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=db=bd, eae-1=ab=ba, ac=ca, ad=da, faf-1=abc, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de3 >

Subgroups: 288 in 128 conjugacy classes, 48 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×2], C4 [×2], C4 [×5], C22 [×3], C22 [×4], C22 [×10], C8 [×3], C2×C4 [×2], C2×C4 [×4], C2×C4 [×19], C23, C23 [×2], C23 [×6], C22⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×5], C22×C4 [×2], C22×C4 [×10], C24, C2.C42 [×2], C22⋊C8 [×2], C2.D8 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×4], C22×C8 [×2], C23×C4, C22.4Q16, C22.4Q16 [×2], C23.7Q8 [×2], C2×C22⋊C8, C2×C2.D8, C24.88D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, Q16 [×2], C2×D4 [×2], C4○D4 [×5], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C2×Q16, C4○D8, C8⋊C22 [×2], C23.11D4, C8.18D4, C82D4, C23.19D4 [×2], C23.48D4 [×2], C24.88D4

Smallest permutation representation of C24.88D4
On 64 points
Generators in S64
(2 9)(4 11)(6 13)(8 15)(17 37)(18 64)(19 39)(20 58)(21 33)(22 60)(23 35)(24 62)(26 50)(28 52)(30 54)(32 56)(34 41)(36 43)(38 45)(40 47)(42 61)(44 63)(46 57)(48 59)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)
(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 24 12 47)(2 23 13 46)(3 22 14 45)(4 21 15 44)(5 20 16 43)(6 19 9 42)(7 18 10 41)(8 17 11 48)(25 60 53 38)(26 59 54 37)(27 58 55 36)(28 57 56 35)(29 64 49 34)(30 63 50 33)(31 62 51 40)(32 61 52 39)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111112222224
type++++++++-+
imageC1C2C2C2C2D4D4D4C4○D4Q16C4○D8C8⋊C22
kernelC24.88D4C22.4Q16C23.7Q8C2×C22⋊C8C2×C2.D8C2×C8C22×C4C24C2×C4C23C22C22
# reps1321121110442

Matrix representation of C24.88D4 in GL6(𝔽17)

100000
0160000
001000
000100
000010
00001516
,
100000
010000
001000
000100
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
1600000
0160000
0016000
0001600
000010
000001
,
1500000
080000
002000
000900
000022
0000615
,
080000
200000
000900
0015000
000088
0000119

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,15,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,0,0,0,0,0,0,8,0,0,0,0,0,0,2,0,0,0,0,0,0,9,0,0,0,0,0,0,2,6,0,0,0,0,2,15],[0,2,0,0,0,0,8,0,0,0,0,0,0,0,0,15,0,0,0,0,9,0,0,0,0,0,0,0,8,11,0,0,0,0,8,9] >;

C24.88D4 in GAP, Magma, Sage, TeX

C_2^4._{88}D_4
% in TeX

G:=Group("C2^4.88D4");
// GroupNames label

G:=SmallGroup(128,808);
// by ID

G=gap.SmallGroup(128,808);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,394,718,172]);
// Polycyclic

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

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