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G = C23.17C42order 128 = 27

12nd non-split extension by C23 of C42 acting via C42/C22=C22

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

Aliases: C23.17C42, C22⋊C813C4, C86(C22⋊C4), (C2×C8).382D4, C4.160(C4×D4), (C2×C4)⋊7M4(2), C2.1(C86D4), C2.2(C89D4), (C2×C4).24C42, C24.50(C2×C4), C22.85(C4×D4), (C2×M4(2))⋊18C4, C2.11(C4×M4(2)), C22.24(C8○D4), C22.52(C2×C42), C4.72(C42⋊C2), C2.C42.14C4, (C23×C4).225C22, C23.255(C22×C4), (C2×C42).991C22, (C22×C8).471C22, C22.39(C2×M4(2)), C2.14(C82M4(2)), (C22×C4).1608C23, C22.7C4238C2, (C22×M4(2)).18C2, (C2×C4×C8)⋊37C2, (C2×C8⋊C4)⋊20C2, (C2×C8).132(C2×C4), C2.11(C4×C22⋊C4), (C2×C4).1498(C2×D4), (C4×C22⋊C4).11C2, (C2×C22⋊C8).43C2, (C2×C22⋊C4).24C4, C4.108(C2×C22⋊C4), (C2×C4).918(C4○D4), (C2×C4).598(C22×C4), (C22×C4).108(C2×C4), SmallGroup(128,485)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.17C42
C1C2C22C2×C4C22×C4C22×C8C2×C4×C8 — C23.17C42
C1C22 — C23.17C42
C1C22×C4 — C23.17C42
C1C2C2C22×C4 — C23.17C42

Generators and relations for C23.17C42
 G = < a,b,c,d,e | a2=b2=c2=d4=1, e4=c, dad-1=ab=ba, eae-1=ac=ca, bc=cb, ede-1=bd=db, be=eb, cd=dc, ce=ec >

Subgroups: 276 in 174 conjugacy classes, 88 normal (36 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×6], C2×C4 [×20], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×4], C2×C8 [×12], C2×C8 [×6], M4(2) [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C2.C42 [×2], C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C22×C8 [×4], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C22.7C42 [×2], C4×C22⋊C4, C2×C4×C8, C2×C8⋊C4, C2×C22⋊C8, C22×M4(2), C23.17C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], M4(2) [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C2×M4(2) [×2], C8○D4 [×2], C4×C22⋊C4, C4×M4(2), C82M4(2), C89D4 [×2], C86D4 [×2], C23.17C42

Smallest permutation representation of C23.17C42
On 64 points
Generators in S64
(2 6)(4 8)(9 13)(11 15)(17 59)(18 64)(19 61)(20 58)(21 63)(22 60)(23 57)(24 62)(26 30)(28 32)(33 56)(34 53)(35 50)(36 55)(37 52)(38 49)(39 54)(40 51)(41 45)(43 47)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 56)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)
(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 21 31 50)(2 64 32 36)(3 23 25 52)(4 58 26 38)(5 17 27 54)(6 60 28 40)(7 19 29 56)(8 62 30 34)(9 22 47 51)(10 57 48 37)(11 24 41 53)(12 59 42 39)(13 18 43 55)(14 61 44 33)(15 20 45 49)(16 63 46 35)
(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)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4P4Q···4V8A···8P8Q···8X
order12···2224···44···44···48···88···8
size11···1441···12···24···42···24···4

56 irreducible representations

dim111111111112222
type++++++++
imageC1C2C2C2C2C2C2C4C4C4C4D4M4(2)C4○D4C8○D4
kernelC23.17C42C22.7C42C4×C22⋊C4C2×C4×C8C2×C8⋊C4C2×C22⋊C8C22×M4(2)C2.C42C22⋊C8C2×C22⋊C4C2×M4(2)C2×C8C2×C4C2×C4C22
# reps121111148484848

Matrix representation of C23.17C42 in GL6(𝔽17)

100000
4160000
001000
0001600
000010
0000016
,
1600000
0160000
0016000
0001600
000010
000001
,
100000
010000
001000
000100
0000160
0000016
,
4150000
0130000
000100
001000
0000160
0000016
,
400000
16130000
004000
0001300
0000016
000040

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

C23.17C42 in GAP, Magma, Sage, TeX

C_2^3._{17}C_4^2
% in TeX

G:=Group("C2^3.17C4^2");
// GroupNames label

G:=SmallGroup(128,485);
// by ID

G=gap.SmallGroup(128,485);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,100,172]);
// Polycyclic

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

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