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G = C24.198C23  order 128 = 27

38th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.198C23, C23.207C24, C22.462+ 1+4, C22.302- 1+4, C4⋊D4⋊17C4, (C22×C4)⋊12D4, C22.1(C4×D4), C23.366(C2×D4), C23.8Q8⋊8C2, C23.23D4⋊7C2, C2.5(C23⋊3D4), C22.98(C23×C4), C23.11(C22×C4), (C23×C4).46C22, C23.7Q8⋊17C2, C22.95(C22×D4), C23.225(C4○D4), (C22×C4).472C23, C24.C22⋊7C2, (C2×C42).414C22, C23.63C23⋊6C2, C2.4(C22.32C24), (C22×D4).478C22, C2.15(C22.11C24), C2.C42.43C22, C2.5(C22.31C24), C2.4(C22.33C24), C2.14(C23.33C23), (C2×C4×D4)⋊7C2, C2.24(C2×C4×D4), C4⋊C4⋊10(C2×C4), (C2×D4)⋊15(C2×C4), C22⋊C4⋊11(C2×C4), (C22×C4)⋊25(C2×C4), (C2×C4⋊D4).16C2, (C2×C4).1187(C2×D4), (C2×C4).28(C22×C4), C22.92(C2×C4○D4), (C2×C4⋊C4).179C22, (C2×C2.C42)⋊17C2, (C2×C22⋊C4).428C22, SmallGroup(128,1057)

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

Derived series
C1 — C22 — C24.198C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42 — C24.198C23
Lower central
C1 — C22 — C24.198C23
Upper central
C1 — C23 — C24.198C23
Jennings
C1 — C23 — C24.198C23

Generators and relations for C24.198C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=e2=c, f2=b, 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: 668 in 348 conjugacy classes, 148 normal (30 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C23×C4, C23×C4, C22×D4, C22×D4, C2×C2.C42, C23.7Q8, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C2×C4×D4, C2×C4⋊D4, C24.198C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4×D4, C22.11C24, C23.33C23, C23⋊3D4, C22.31C24, C22.32C24, C22.33C24, C24.198C23

Smallest permutation representation of C24.198C23
►On 64 points
Generators in S64
(1 12)(2 9)(3 10)(4 11)(5 38)(6 39)(7 40)(8 37)(13 36)(14 33)(15 34)(16 35)(17 60)(18 57)(19 58)(20 59)(21 56)(22 53)(23 54)(24 55)(25 48)(26 45)(27 46)(28 47)(29 52)(30 49)(31 50)(32 51)(41 64)(42 61)(43 62)(44 63)

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I···4AB
order12···2222222224···44···4
size11···1222244442···24···4

44 irreducible representations

dim11111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C4D4C4○D42+ 1+42- 1+4
kernelC24.198C23C2×C2.C42C23.7Q8C23.8Q8C23.23D4C23.63C23C24.C22C2×C4×D4C2×C4⋊D4C4⋊D4C22×C4C23C22C22
# reps111242221164431

Matrix representation of C24.198C23 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
30000000
12000000
00400000
00010000
00003000
00000200
00000020
00000003
,
30000000
03000000
00400000
00040000
00000200
00002000
00000003
00000030
,
41000000
31000000
00010000
00400000
00000010
00000001
00004000
00000400
,
40000000
04000000
00100000
00010000
00000100
00001000
00000001
00000010

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

C24.198C23 in GAP, Magma, Sage, TeX 

C_2^4._{198}C_2^3
 
% in TeX
 
G:=Group("C2^4.198C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1057);
 
// by ID
 
G=gap.SmallGroup(128,1057);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,184,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=e^2=c,f^2=b,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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