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G = C23.199C24  order 128 = 27

52nd central extension by C23 of C24

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

Aliases: C23.199C24, C24.546C23, C22.382+ 1+4, C22.232- 1+4, C42⋊16(C2×C4), C42⋊9C4⋊10C2, C42⋊C2⋊25C4, C42⋊8C4⋊11C2, (C22×C4).54Q8, C23.93(C2×Q8), C23.605(C2×D4), (C22×C4).362D4, (C23×C4).43C22, C22.90(C23×C4), C22.90(C22×D4), C23.8Q8.3C2, C22.32(C22×Q8), (C2×C42).408C22, (C22×C4).464C23, C23.123(C22×C4), C2.3(C22.29C24), C23.65C23⋊10C2, C2.C42.36C22, C2.3(C23.38C23), C2.7(C23.33C23), C2.2(C23.41C23), C4⋊C4⋊40(C2×C4), (C2×C4)⋊6(C4⋊C4), C4.59(C2×C4⋊C4), (C2×C4).829(C2×D4), C22.31(C2×C4⋊C4), C2.13(C22×C4⋊C4), (C2×C4).229(C2×Q8), (C22×C4⋊C4).26C2, C22⋊C4.57(C2×C4), (C2×C4⋊C4).173C22, (C22×C4).301(C2×C4), (C2×C4).222(C22×C4), (C2×C42⋊C2).28C2, (C2×C22⋊C4).482C22, SmallGroup(128,1049)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C23.199C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C42⋊C2 — C23.199C24
Lower central
C1 — C22 — C23.199C24
Upper central
C1 — C23 — C23.199C24
Jennings
C1 — C23 — C23.199C24

Generators and relations for C23.199C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=c, f2=b, g2=a, ab=ba, ac=ca, ede=gdg-1=ad=da, fef-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 476 in 300 conjugacy classes, 180 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C23×C4, C23×C4, C42⋊8C4, C42⋊9C4, C23.8Q8, C23.65C23, C22×C4⋊C4, C2×C42⋊C2, C23.199C24
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, 2+ 1+4, 2- 1+4, C22×C4⋊C4, C23.33C23, C22.29C24, C23.38C23, C23.41C23, C23.199C24

Smallest permutation representation of C23.199C24
►On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AF
order12···222224···44···4
size11···122222···24···4

44 irreducible representations

dim111111112244
type++++++++-+-
imageC1C2C2C2C2C2C2C4D4Q82+ 1+42- 1+4
kernelC23.199C24C42⋊8C4C42⋊9C4C23.8Q8C23.65C23C22×C4⋊C4C2×C42⋊C2C42⋊C2C22×C4C22×C4C22C22
# reps1224412164422

Matrix representation of C23.199C24 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
02000000
20000000
00040000
00400000
00000040
00000114
00004000
00004014
,
10000000
01000000
00400000
00040000
00001000
00000100
00000040
00000204
,
20000000
03000000
00300000
00020000
00000010
00000144
00001000
00004044
,
40000000
04000000
00100000
00010000
00004400
00002100
00000401
00002140

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

C23.199C24 in GAP, Magma, Sage, TeX 

C_2^3._{199}C_2^4
 
% in TeX
 
G:=Group("C2^3.199C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1049);
 
// by ID
 
G=gap.SmallGroup(128,1049);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,219,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=c,f^2=b,g^2=a,a*b=b*a,a*c=c*a,e*d*e=g*d*g^-1=a*d=d*a,f*e*f^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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