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G = C24.76D4order 128 = 27

31st non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.76D4, (C2×C8)⋊23D4, C4⋊D413C4, C4.115(C4×D4), C4.92C22≀C2, C2.3(C8⋊D4), C2.2(C82D4), C23.775(C2×D4), (C22×C4).291D4, C23.7Q87C2, C22.4Q1648C2, C23.82(C22⋊C4), C22.80(C8⋊C22), (C22×M4(2))⋊11C2, (C22×C8).390C22, (C23×C4).260C22, (C22×D4).25C22, C22.121(C4⋊D4), (C22×C4).1371C23, C4.87(C22.D4), C22.69(C8.C22), C2.34(C23.23D4), C2.27(C23.36D4), C2.21(C23.37D4), C4⋊C4.74(C2×C4), (C2×D4).82(C2×C4), (C2×C4⋊D4).9C2, (C2×D4⋊C4)⋊44C2, (C2×C4).1334(C2×D4), (C2×C4⋊C4).63C22, (C2×C4).568(C4○D4), (C22×C4).281(C2×C4), (C2×C4).389(C22×C4), (C2×C4).133(C22⋊C4), C22.270(C2×C22⋊C4), SmallGroup(128,627)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.76D4
C1C2C22C23C22×C4C23×C4C22×M4(2) — C24.76D4
C1C2C2×C4 — C24.76D4
C1C23C23×C4 — C24.76D4
C1C2C2C22×C4 — C24.76D4

Generators and relations for C24.76D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, eae-1=ad=da, faf=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >

Subgroups: 484 in 206 conjugacy classes, 64 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22 [×3], C22 [×4], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×14], C23, C23 [×2], C23 [×14], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C2×C8 [×4], M4(2) [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×7], C2×D4 [×2], C2×D4 [×13], C24, C24, C2.C42, D4⋊C4 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C22×C8 [×2], C2×M4(2) [×6], C23×C4, C22×D4, C22×D4, C22.4Q16 [×2], C23.7Q8, C2×D4⋊C4 [×2], C2×C4⋊D4, C22×M4(2), C24.76D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C8⋊C22 [×3], C8.C22, C23.23D4, C23.36D4, C23.37D4, C8⋊D4 [×2], C82D4 [×2], C24.76D4

Smallest permutation representation of C24.76D4
On 64 points
Generators in S64
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 58)(10 63)(11 60)(12 57)(13 62)(14 59)(15 64)(16 61)(17 54)(18 51)(19 56)(20 53)(21 50)(22 55)(23 52)(24 49)(33 42)(34 47)(35 44)(36 41)(37 46)(38 43)(39 48)(40 45)
(1 37)(2 38)(3 39)(4 40)(5 33)(6 34)(7 35)(8 36)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 49)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)
(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 56)(2 6)(3 54)(5 52)(7 50)(9 22)(10 31)(11 20)(12 29)(13 18)(14 27)(15 24)(16 25)(17 44)(19 42)(21 48)(23 46)(26 45)(28 43)(30 41)(32 47)(33 63)(34 38)(35 61)(37 59)(39 57)(51 55)(58 62)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G···4L8A···8H
order12···222224444444···48···8
size11···144882222448···84···4

32 irreducible representations

dim1111111222244
type++++++++++-
imageC1C2C2C2C2C2C4D4D4D4C4○D4C8⋊C22C8.C22
kernelC24.76D4C22.4Q16C23.7Q8C2×D4⋊C4C2×C4⋊D4C22×M4(2)C4⋊D4C2×C8C22×C4C24C2×C4C22C22
# reps1212118431431

Matrix representation of C24.76D4 in GL8(𝔽17)

169000000
01000000
00100000
00010000
00000010
00000001
00001000
00000100
,
160000000
016000000
00100000
00010000
00001000
00000100
00000010
00000001
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
130000000
013000000
001120000
00860000
000004134
0000154150
0000413013
000020213
,
10000000
416000000
001600000
001110000
000016200
00000100
000000115
000000016

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,11,8,0,0,0,0,0,0,2,6,0,0,0,0,0,0,0,0,0,15,4,2,0,0,0,0,4,4,13,0,0,0,0,0,13,15,0,2,0,0,0,0,4,0,13,13],[1,4,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,15,16] >;

C24.76D4 in GAP, Magma, Sage, TeX

C_2^4._{76}D_4
% in TeX

G:=Group("C2^4.76D4");
// GroupNames label

G:=SmallGroup(128,627);
// by ID

G=gap.SmallGroup(128,627);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,723,2019,248]);
// Polycyclic

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

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