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G = C24.83D4order 128 = 27

38th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.83D4, C23.11D8, (C2×C8)⋊7D4, C4⋊C4.88D4, (C2×D4).99D4, C22.83(C2×D8), C2.17(C8⋊D4), C2.13(C87D4), C23.909(C2×D4), (C22×C4).145D4, C2.21(C22⋊D8), C23.7Q89C2, C4.142(C4⋊D4), C22.4Q1621C2, C4.36(C4.4D4), (C22×C8).69C22, C22.215C22≀C2, C2.31(D4.7D4), C22.107(C4○D8), (C23×C4).271C22, C2.6(C22.D8), (C22×D4).76C22, C22.226(C4⋊D4), C22.135(C8⋊C22), (C22×C4).1443C23, C4.17(C22.D4), C2.8(C23.19D4), C2.6(C23.10D4), C22.124(C8.C22), C22.112(C22.D4), (C2×C2.D8)⋊7C2, (C2×C22⋊C8)⋊19C2, (C2×D4⋊C4)⋊12C2, (C2×C4).1035(C2×D4), (C2×C4⋊D4).13C2, (C2×C4).770(C4○D4), (C2×C4⋊C4).118C22, SmallGroup(128,765)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C24.83D4
C1C2C22C2×C4C22×C4C2×C4⋊C4C23.7Q8 — C24.83D4
C1C2C22×C4 — C24.83D4
C1C23C23×C4 — C24.83D4
C1C2C2C22×C4 — C24.83D4

Generators and relations for C24.83D4
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, eae=ab=ba, ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=bd-1 >

Subgroups: 464 in 184 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×5], C22 [×7], C22 [×20], C8 [×3], C2×C4 [×6], C2×C4 [×17], D4 [×14], C23, C23 [×2], C23 [×14], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×5], C22×C4 [×2], C22×C4 [×9], C2×D4 [×2], C2×D4 [×13], C24, C24, C2.C42, C22⋊C8 [×2], D4⋊C4 [×4], C2.D8 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C4⋊D4 [×4], C22×C8 [×2], C23×C4, C22×D4, C22×D4, C22.4Q16, C23.7Q8, C2×C22⋊C8, C2×D4⋊C4 [×2], C2×C2.D8, C2×C4⋊D4, C24.83D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, D8 [×2], C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C2×D8, C4○D8, C8⋊C22, C8.C22, C23.10D4, C22⋊D8, D4.7D4, C87D4, C8⋊D4, C22.D8, C23.19D4, C24.83D4

Smallest permutation representation of C24.83D4
On 64 points
Generators in S64
(1 26)(2 63)(3 28)(4 57)(5 30)(6 59)(7 32)(8 61)(9 52)(10 46)(11 54)(12 48)(13 56)(14 42)(15 50)(16 44)(17 47)(18 55)(19 41)(20 49)(21 43)(22 51)(23 45)(24 53)(25 38)(27 40)(29 34)(31 36)(33 64)(35 58)(37 60)(39 62)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 41)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)
(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)
(2 41)(3 7)(4 47)(6 45)(8 43)(9 11)(10 30)(12 28)(13 15)(14 26)(16 32)(17 23)(18 64)(19 21)(20 62)(22 60)(24 58)(25 27)(29 31)(33 37)(34 54)(36 52)(38 50)(40 56)(44 48)(51 55)(57 59)(61 63)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111112222222244
type++++++++++++++-
imageC1C2C2C2C2C2C2D4D4D4D4D4C4○D4D8C4○D8C8⋊C22C8.C22
kernelC24.83D4C22.4Q16C23.7Q8C2×C22⋊C8C2×D4⋊C4C2×C2.D8C2×C4⋊D4C4⋊C4C2×C8C22×C4C2×D4C24C2×C4C23C22C22C22
# reps11112112212164411

Matrix representation of C24.83D4 in GL6(𝔽17)

16160000
010000
000100
001000
000010
000001
,
1600000
0160000
0016000
0001600
000010
000001
,
1600000
0160000
001000
000100
000010
000001
,
1300000
0130000
004000
0001300
0000011
0000311
,
100000
15160000
001000
0001600
000010
0000116

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,16,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[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],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,3,0,0,0,0,11,11],[1,15,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;

C24.83D4 in GAP, Magma, Sage, TeX

C_2^4._{83}D_4
% in TeX

G:=Group("C2^4.83D4");
// GroupNames label

G:=SmallGroup(128,765);
// by ID

G=gap.SmallGroup(128,765);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,e*a*e=a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b*d^-1>;
// generators/relations

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