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G = C24.50D4order 128 = 27

5th non-split extension by C24 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C24.50D4, C25.3C22, C24.628C23, (C22×C4)⋊9D4, (C22×D4)⋊14C4, C24.41(C2×C4), (D4×C23).1C2, C234(C22⋊C4), C22.121(C4×D4), C23.718(C2×D4), (C23×C4).6C22, C2.3(C243C4), C22.65C22≀C2, C2.1(C232D4), C23.341(C4○D4), C22.23(C41D4), C22.97(C4⋊D4), C23.297(C22×C4), C2.5(C23.23D4), C2.1(C23.10D4), C22.47(C4.4D4), C2.4(C24.3C22), C22.70(C22.D4), (C2×C4)⋊4(C22⋊C4), (C22×C22⋊C4)⋊1C2, (C22×C4).167(C2×C4), (C2×C2.C42)⋊12C2, C22.138(C2×C22⋊C4), SmallGroup(128,170)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.50D4
C1C2C22C23C24C25D4×C23 — C24.50D4
C1C23 — C24.50D4
C1C24 — C24.50D4
C1C24 — C24.50D4

Generators and relations for C24.50D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, 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=ce-1 >

Subgroups: 1268 in 584 conjugacy classes, 124 normal (8 characteristic)
C1, C2, C2 [×14], C2 [×8], C4 [×12], C22, C22 [×34], C22 [×72], C2×C4 [×4], C2×C4 [×52], D4 [×32], C23, C23 [×22], C23 [×104], C22⋊C4 [×32], C22×C4 [×14], C22×C4 [×28], C2×D4 [×56], C24, C24 [×12], C24 [×24], C2.C42 [×4], C2×C22⋊C4 [×24], C23×C4, C23×C4 [×4], C22×D4 [×4], C22×D4 [×12], C25 [×2], C2×C2.C42 [×2], C22×C22⋊C4 [×4], D4×C23, C24.50D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×20], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×10], C4○D4 [×4], C2×C22⋊C4 [×3], C4×D4 [×4], C22≀C2 [×8], C4⋊D4 [×12], C22.D4 [×4], C4.4D4 [×2], C41D4 [×2], C243C4, C23.23D4 [×4], C24.3C22 [×2], C232D4 [×4], C23.10D4 [×4], C24.50D4

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

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

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

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

44 conjugacy classes

class 1 2A···2O2P···2W4A···4T
order12···22···24···4
size11···14···44···4

44 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D4D4C4○D4
kernelC24.50D4C2×C2.C42C22×C22⋊C4D4×C23C22×D4C22×C4C24C23
# reps124181288

Matrix representation of C24.50D4 in GL8(𝔽5)

04000000
40000000
00100000
00010000
00001400
00000400
00000010
00000024
,
10000000
01000000
00100000
00010000
00001000
00000100
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00100000
00010000
00004000
00000400
00000010
00000001
,
30000000
03000000
00220000
00130000
00004000
00000400
00000041
00000031
,
20000000
03000000
00220000
00030000
00004000
00003100
00000014
00000004

G:=sub<GL(8,GF(5))| [0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,1,1],[2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,4] >;

C24.50D4 in GAP, Magma, Sage, TeX

C_2^4._{50}D_4
% in TeX

G:=Group("C2^4.50D4");
// GroupNames label

G:=SmallGroup(128,170);
// by ID

G=gap.SmallGroup(128,170);
# by ID

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

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

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