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G = C24.65D4order 128 = 27

20th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.65D4, C4.54C22≀C2, D44(C22⋊C4), (C2×D4).264D4, Q84(C22⋊C4), (C2×Q8).207D4, C2.1(D4⋊D4), (C22×C4).675D4, C23.739(C2×D4), (C22×C8).7C22, C23.7Q84C2, C22.74C22≀C2, C2.1(D4.7D4), C22.40(C4○D8), C2.13(C243C4), C23.72(C22⋊C4), C22.58(C8⋊C22), (C23×C4).233C22, (C22×C4).1322C23, (C22×D4).453C22, C22.47(C8.C22), (C22×Q8).381C22, C2.20(C23.36D4), C2.20(C23.24D4), (C2×C4○D4)⋊11C4, C4.3(C2×C22⋊C4), (C2×D4⋊C4)⋊2C2, (C2×C22⋊C8)⋊12C2, (C2×Q8⋊C4)⋊2C2, (C2×C4).973(C2×D4), (C2×D4).201(C2×C4), (C2×C4⋊C4).30C22, (C2×Q8).184(C2×C4), (C22×C4○D4).4C2, (C2×C4).360(C22×C4), (C22×C4).262(C2×C4), (C2×C4).330(C22⋊C4), C22.241(C2×C22⋊C4), SmallGroup(128,520)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.65D4
C1C2C22C23C22×C4C23×C4C22×C4○D4 — C24.65D4
C1C2C2×C4 — C24.65D4
C1C23C23×C4 — C24.65D4
C1C2C2C22×C4 — C24.65D4

Generators and relations for C24.65D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, eae-1=ab=ba, ac=ca, ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

Subgroups: 620 in 306 conjugacy classes, 76 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22 [×3], C22 [×4], C22 [×26], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×38], D4 [×4], D4 [×22], Q8 [×4], Q8 [×6], C23, C23 [×2], C23 [×16], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×19], C2×D4 [×6], C2×D4 [×15], C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×32], C24, C24, C2.C42, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C2×C22⋊C4, C2×C4⋊C4 [×2], C22×C8 [×2], C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4 [×4], C2×C4○D4 [×10], C23.7Q8, C2×C22⋊C8, C2×D4⋊C4 [×2], C2×Q8⋊C4 [×2], C22×C4○D4, C24.65D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×6], C2×C22⋊C4 [×3], C22≀C2 [×4], C4○D8 [×2], C8⋊C22, C8.C22, C243C4, C23.24D4, C23.36D4, D4⋊D4 [×2], D4.7D4 [×2], C24.65D4

Smallest permutation representation of C24.65D4
On 64 points
Generators in S64
(2 63)(4 57)(6 59)(8 61)(9 21)(10 14)(11 23)(12 16)(13 17)(15 19)(18 22)(20 24)(25 48)(26 30)(27 42)(28 32)(29 44)(31 46)(33 55)(35 49)(37 51)(39 53)(41 45)(43 47)
(1 62)(2 63)(3 64)(4 57)(5 58)(6 59)(7 60)(8 61)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)(33 55)(34 56)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)
(1 40)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 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 25 40 17)(2 24 33 32)(3 31 34 23)(4 22 35 30)(5 29 36 21)(6 20 37 28)(7 27 38 19)(8 18 39 26)(9 62 44 54)(10 53 45 61)(11 60 46 52)(12 51 47 59)(13 58 48 50)(14 49 41 57)(15 64 42 56)(16 55 43 63)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2M4A···4H4I4J4K4L4M4N4O4P8A···8H
order12···22···24···4444444448···8
size11···14···42···2444488884···4

38 irreducible representations

dim11111112222244
type+++++++++++-
imageC1C2C2C2C2C2C4D4D4D4D4C4○D8C8⋊C22C8.C22
kernelC24.65D4C23.7Q8C2×C22⋊C8C2×D4⋊C4C2×Q8⋊C4C22×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C24C22C22C22
# reps11122183441811

Matrix representation of C24.65D4 in GL5(𝔽17)

160000
01000
001600
00010
000816
,
10000
01000
00100
000160
000016
,
160000
01000
00100
00010
00001
,
10000
016000
001600
00010
00001
,
130000
015000
00800
000815
00079
,
130000
00800
015000
00092
000118

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1,8,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[13,0,0,0,0,0,15,0,0,0,0,0,8,0,0,0,0,0,8,7,0,0,0,15,9],[13,0,0,0,0,0,0,15,0,0,0,8,0,0,0,0,0,0,9,11,0,0,0,2,8] >;

C24.65D4 in GAP, Magma, Sage, TeX

C_2^4._{65}D_4
% in TeX

G:=Group("C2^4.65D4");
// GroupNames label

G:=SmallGroup(128,520);
// by ID

G=gap.SmallGroup(128,520);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,352,2019,1018,248]);
// Polycyclic

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

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