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G = C24.361C23  order 128 = 27

201st non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.361C23, C23.516C24, C22.2152- 1+4, C22.2942+ 1+4, C4.104C22≀C2, (C22×C4).398D4, C23.189(C2×D4), C23.7Q8⋊75C2, C23.10D4⋊53C2, (C23×C4).419C22, (C22×C4).854C23, C22.341(C22×D4), (C22×D4).539C22, (C22×Q8).448C22, C23.78C23⋊25C2, C2.C42.244C22, C2.32(C23.38C23), C2.22(C22.31C24), (C2×C4).376(C2×D4), (C2×C22⋊Q8)⋊26C2, C2.28(C2×C22≀C2), (C2×C4⋊C4).354C22, (C22×C4○D4).12C2, (C2×C22⋊C4).209C22, SmallGroup(128,1348)

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

Derived series
C1 — C23 — C24.361C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — C2×C22⋊Q8 — C24.361C23
Lower central
C1 — C23 — C24.361C23
Upper central
C1 — C23 — C24.361C23
Jennings
C1 — C23 — C24.361C23

Generators and relations for C24.361C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=g2=b, eae=ab=ba, faf-1=ac=ca, ad=da, ag=ga, bc=cb, bd=db, be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 772 in 408 conjugacy classes, 116 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C23.7Q8, C23.10D4, C23.78C23, C2×C22⋊Q8, C22×C4○D4, C24.361C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, 2+ 1+4, 2- 1+4, C2×C22≀C2, C23.38C23, C22.31C24, C24.361C23

Smallest permutation representation of C24.361C23
►On 64 points
Generators in S64
(1 6)(2 21)(3 8)(4 23)(5 12)(7 10)(9 24)(11 22)(13 18)(14 64)(15 20)(16 62)(17 28)(19 26)(25 63)(27 61)(29 60)(30 46)(31 58)(32 48)(33 59)(34 45)(35 57)(36 47)(37 53)(38 43)(39 55)(40 41)(42 52)(44 50)(49 54)(51 56)

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H···2M4A4B4C4D4E···4J4K···4R
order12···22···244444···44···4
size11···14···422224···48···8

32 irreducible representations

dim111111244
type++++++++-
imageC1C2C2C2C2C2D42+ 1+42- 1+4
kernelC24.361C23C23.7Q8C23.10D4C23.78C23C2×C22⋊Q8C22×C4○D4C22×C4C22C22
# reps1362311213

Matrix representation of C24.361C23 ►in GL8(𝔽5)

10000000
01000000
00100000
00040000
00003040
00000204
00003020
00000303
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
01000000
00400000
00010000
00001030
00000403
00000040
00000001
,
04000000
40000000
00040000
00400000
00000100
00004000
00000004
00000010
,
40000000
04000000
00100000
00010000
00003000
00000200
00000030
00000002

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

C24.361C23 in GAP, Magma, Sage, TeX 

C_2^4._{361}C_2^3
 
% in TeX
 
G:=Group("C2^4.361C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1348);
 
// by ID
 
G=gap.SmallGroup(128,1348);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,100,185,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=1,f^2=g^2=b,e*a*e=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,g*f*g^-1=b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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