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G = C23.192C24order 128 = 27

45th central extension by C23 of C24

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

Aliases: C23.192C24, C24.541C23, C22.182- 1+4, (C22×Q8)⋊17C4, (Q8×C23).5C2, C23.603(C2×D4), (C22×C4).359D4, C22.83(C23×C4), C22.86(C22×D4), (C23×C4).286C22, (C22×C4).457C23, C23.210(C22×C4), (C2×C42).405C22, C23.34D4.7C2, (C22×Q8).395C22, C23.67C2311C2, C2.C42.31C22, C2.1(C23.38C23), C2.4(C23.32C23), (C2×C4).828(C2×D4), C4.64(C2×C22⋊C4), (C2×Q8).192(C2×C4), (C2×C4⋊C4).804C22, (C2×C4).215(C22×C4), (C22×C4).298(C2×C4), C2.14(C22×C22⋊C4), C22.75(C2×C22⋊C4), (C2×C4).154(C22⋊C4), (C2×C42⋊C2).25C2, (C2×C22⋊C4).422C22, SmallGroup(128,1042)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.192C24
C1C2C22C23C24C23×C4Q8×C23 — C23.192C24
C1C22 — C23.192C24
C1C23 — C23.192C24
C1C23 — C23.192C24

Generators and relations for C23.192C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=c, e2=f2=a, ab=ba, ac=ca, ede-1=gdg=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: 620 in 392 conjugacy classes, 180 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C4 [×16], C22, C22 [×10], C22 [×12], C2×C4 [×36], C2×C4 [×48], Q8 [×32], C23, C23 [×6], C23 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×34], C22×C4 [×8], C2×Q8 [×16], C2×Q8 [×48], C24, C2.C42 [×16], C2×C42 [×4], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C42⋊C2 [×8], C23×C4, C23×C4 [×2], C22×Q8 [×12], C22×Q8 [×8], C23.34D4 [×4], C23.67C23 [×8], C2×C42⋊C2 [×2], Q8×C23, C23.192C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], 2- 1+4 [×4], C22×C22⋊C4, C23.32C23 [×2], C23.38C23 [×4], C23.192C24

Smallest permutation representation of C23.192C24
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 17 9 45)(2 46 10 18)(3 19 11 47)(4 48 12 20)(5 16 38 44)(6 41 39 13)(7 14 40 42)(8 43 37 15)(21 57 49 29)(22 30 50 58)(23 59 51 31)(24 32 52 60)(25 61 53 35)(26 36 54 62)(27 63 55 33)(28 34 56 64)
(1 13 9 41)(2 28 10 56)(3 15 11 43)(4 26 12 54)(5 30 38 58)(6 17 39 45)(7 32 40 60)(8 19 37 47)(14 24 42 52)(16 22 44 50)(18 34 46 64)(20 36 48 62)(21 53 49 25)(23 55 51 27)(29 35 57 61)(31 33 59 63)
(1 3)(2 12)(4 10)(5 40)(6 8)(7 38)(9 11)(13 15)(14 44)(16 42)(17 19)(18 48)(20 46)(21 23)(22 52)(24 50)(25 27)(26 56)(28 54)(29 31)(30 60)(32 58)(33 35)(34 62)(36 64)(37 39)(41 43)(45 47)(49 51)(53 55)(57 59)(61 63)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim11111124
type++++++-
imageC1C2C2C2C2C4D42- 1+4
kernelC23.192C24C23.34D4C23.67C23C2×C42⋊C2Q8×C23C22×Q8C22×C4C22
# reps148211684

Matrix representation of C23.192C24 in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
03000000
20000000
00040000
00100000
00004030
00004403
00000010
00001011
,
10000000
01000000
00400000
00040000
00002200
00000300
00002333
00000102
,
40000000
01000000
00100000
00040000
00003000
00004200
00000030
00002042
,
40000000
04000000
00100000
00010000
00001000
00000100
00004040
00004404

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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,4,0,1,0,0,0,0,0,4,0,0,0,0,0,0,3,0,1,1,0,0,0,0,0,3,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0,2,3,3,1,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2],[4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,4,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,4,4,0,0,0,0,0,1,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4] >;

C23.192C24 in GAP, Magma, Sage, TeX

C_2^3._{192}C_2^4
% in TeX

G:=Group("C2^3.192C2^4");
// GroupNames label

G:=SmallGroup(128,1042);
// by ID

G=gap.SmallGroup(128,1042);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,219,100,675]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c,e^2=f^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g=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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