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G = C24.17Q8order 128 = 27

1st non-split extension by C24 of Q8 acting via Q8/C4=C2

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

Aliases: C24.17Q8, C24.147D4, C23.23C42, C25.80C22, C24.623C23, (C23×C4)⋊12C4, (C24×C4).1C2, C24.91(C2×C4), C22.72(C4×D4), C23.46(C4⋊C4), C23.713(C2×D4), (C22×C4).644D4, (C23×C4).3C22, C23.125(C2×Q8), C2.1(C243C4), C22.62C22≀C2, C22⋊(C2.C42), C22.35(C2×C42), C23.336(C4○D4), C22.92(C4⋊D4), C23.89(C22⋊C4), C23.237(C22×C4), C22.56(C22⋊Q8), C2.1(C23.8Q8), C2.1(C23.7Q8), C2.1(C23.34D4), C2.1(C23.23D4), C22.40(C42⋊C2), C22.66(C22.D4), (C22×C4)⋊9(C2×C4), C2.4(C4×C22⋊C4), (C2×C22⋊C4)⋊10C4, C22.41(C2×C4⋊C4), (C2×C4)⋊11(C22⋊C4), (C2×C2.C42)⋊1C2, (C22×C22⋊C4).1C2, C22.83(C2×C22⋊C4), C2.3(C2×C2.C42), SmallGroup(128,165)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.17Q8
C1C2C22C23C24C25C24×C4 — C24.17Q8
C1C22 — C24.17Q8
C1C24 — C24.17Q8
C1C24 — C24.17Q8

Generators and relations for C24.17Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=de2, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be-1 >

Subgroups: 964 in 544 conjugacy classes, 172 normal (14 characteristic)
C1, C2 [×3], C2 [×12], C2 [×8], C4 [×16], C22 [×3], C22 [×40], C22 [×56], C2×C4 [×8], C2×C4 [×96], C23 [×3], C23 [×40], C23 [×56], C22⋊C4 [×16], C22×C4 [×20], C22×C4 [×88], C24, C24 [×14], C24 [×8], C2.C42 [×8], C2×C22⋊C4 [×8], C2×C22⋊C4 [×8], C23×C4 [×10], C23×C4 [×12], C25, C2×C2.C42 [×4], C22×C22⋊C4 [×2], C24×C4, C24.17Q8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×14], Q8 [×2], C23, C42 [×4], C22⋊C4 [×20], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×7], C2×Q8, C4○D4 [×4], C2.C42 [×8], C2×C42, C2×C22⋊C4 [×5], C2×C4⋊C4 [×3], C42⋊C2 [×2], C4×D4 [×8], C22≀C2 [×4], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C2×C2.C42, C4×C22⋊C4 [×2], C243C4, C23.7Q8 [×2], C23.34D4, C23.8Q8 [×4], C23.23D4 [×4], C24.17Q8

Smallest permutation representation of C24.17Q8
On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 31)(6 32)(7 29)(8 30)(9 15)(10 16)(11 13)(12 14)(17 47)(18 48)(19 45)(20 46)(21 43)(22 44)(23 41)(24 42)(33 60)(34 57)(35 58)(36 59)(37 53)(38 54)(39 55)(40 56)(49 64)(50 61)(51 62)(52 63)
(1 5)(2 6)(3 7)(4 8)(9 23)(10 24)(11 21)(12 22)(13 43)(14 44)(15 41)(16 42)(17 37)(18 38)(19 39)(20 40)(25 31)(26 32)(27 29)(28 30)(33 51)(34 52)(35 49)(36 50)(45 55)(46 56)(47 53)(48 54)(57 63)(58 64)(59 61)(60 62)
(1 11)(2 12)(3 9)(4 10)(5 21)(6 22)(7 23)(8 24)(13 25)(14 26)(15 27)(16 28)(17 63)(18 64)(19 61)(20 62)(29 41)(30 42)(31 43)(32 44)(33 56)(34 53)(35 54)(36 55)(37 57)(38 58)(39 59)(40 60)(45 50)(46 51)(47 52)(48 49)
(1 25)(2 26)(3 27)(4 28)(5 31)(6 32)(7 29)(8 30)(9 15)(10 16)(11 13)(12 14)(17 52)(18 49)(19 50)(20 51)(21 43)(22 44)(23 41)(24 42)(33 40)(34 37)(35 38)(36 39)(45 61)(46 62)(47 63)(48 64)(53 57)(54 58)(55 59)(56 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 63 27 45)(2 60 28 54)(3 61 25 47)(4 58 26 56)(5 57 29 55)(6 62 30 48)(7 59 31 53)(8 64 32 46)(9 19 13 52)(10 38 14 33)(11 17 15 50)(12 40 16 35)(18 44 51 24)(20 42 49 22)(21 37 41 36)(23 39 43 34)

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

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

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

56 conjugacy classes

class 1 2A···2O2P···2W4A···4P4Q···4AF
order12···22···24···44···4
size11···12···22···24···4

56 irreducible representations

dim1111112222
type++++++-
imageC1C2C2C2C4C4D4D4Q8C4○D4
kernelC24.17Q8C2×C2.C42C22×C22⋊C4C24×C4C2×C22⋊C4C23×C4C22×C4C24C24C23
# reps14211688628

Matrix representation of C24.17Q8 in GL6(𝔽5)

400000
010000
004000
000100
000040
000004
,
400000
010000
004000
000400
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
040000
004000
000400
000040
000004
,
300000
040000
002000
000200
000030
000002
,
300000
030000
000100
001000
000001
000010

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C24.17Q8 in GAP, Magma, Sage, TeX

C_2^4._{17}Q_8
% in TeX

G:=Group("C2^4.17Q8");
// GroupNames label

G:=SmallGroup(128,165);
// by ID

G=gap.SmallGroup(128,165);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,2,224,141,456,422]);
// Polycyclic

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

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