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G = C24.378C23  order 128 = 27

218th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.378C23, C23.563C24, C22.3372+ 1+4, C22.2522- 1+4, C23⋊Q8⋊36C2, (C22×C4).409D4, C23.201(C2×D4), C23.4Q8⋊37C2, C23.11D4⋊72C2, C2.34(C23⋊3D4), (C2×C42).627C22, (C22×C4).168C23, (C23×C4).438C22, C22.375(C22×D4), C23.10D4.37C2, (C22×D4).211C22, (C22×Q8).169C22, C23.78C23⋊34C2, C23.83C23⋊71C2, C24.C22⋊113C2, C23.65C23⋊110C2, C2.C42.277C22, C2.52(C22.26C24), C2.49(C23.38C23), C2.64(C22.36C24), C2.53(C22.33C24), (C4×C22⋊C4)⋊99C2, (C2×C4).684(C2×D4), (C2×C22⋊Q8)⋊31C2, (C2×C4).183(C4○D4), (C2×C4⋊C4).385C22, C22.430(C2×C4○D4), (C2×C22⋊C4).521C22, (C2×C22.D4).21C2, SmallGroup(128,1395)

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

Derived series
C1 — C23 — C24.378C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C24.C22 — C24.378C23
Lower central
C1 — C23 — C24.378C23
Upper central
C1 — C23 — C24.378C23
Jennings
C1 — C23 — C24.378C23

Generators and relations for C24.378C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=db=bd, g2=c, eae-1=ab=ba, faf=ac=ca, ad=da, ag=ga, bc=cb, be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef=de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 500 in 249 conjugacy classes, 96 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C22.D4, C23×C4, C22×D4, C22×Q8, C4×C22⋊C4, C24.C22, C23.65C23, C23⋊Q8, C23.10D4, C23.78C23, C23.11D4, C23.4Q8, C23.83C23, C2×C22⋊Q8, C2×C22.D4, C24.378C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.26C24, C23⋊3D4, C23.38C23, C22.33C24, C22.36C24, C24.378C23

Smallest permutation representation of C24.378C23
►On 64 points
Generators in S64
(1 55)(2 45)(3 53)(4 47)(5 35)(6 51)(7 33)(8 49)(9 54)(10 48)(11 56)(12 46)(13 59)(14 41)(15 57)(16 43)(17 39)(18 29)(19 37)(20 31)(21 50)(22 36)(23 52)(24 34)(25 42)(26 58)(27 44)(28 60)(30 62)(32 64)(38 63)(40 61)

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

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

(1 12)(2 9)(3 10)(4 11)(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 38)(30 39)(31 40)(32 37)(33 50)(34 51)(35 52)(36 49)(41 58)(42 59)(43 60)(44 57)(45 54)(46 55)(47 56)(48 53)

(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)

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J4A4B4C4D4E···4N4O···4U
order12···222244444···44···4
size11···144822224···48···8

32 irreducible representations

dim1111111111112244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC24.378C23C4×C22⋊C4C24.C22C23.65C23C23⋊Q8C23.10D4C23.78C23C23.11D4C23.4Q8C23.83C23C2×C22⋊Q8C2×C22.D4C22×C4C2×C4C22C22
# reps1122121211114822

Matrix representation of C24.378C23 ►in GL8(𝔽5)

43000000
01000000
00400000
00040000
00004030
00000422
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00010000
00400000
00003000
00000300
00002020
00003302
,
10000000
44000000
00100000
00040000
00001000
00003400
00000010
00000004
,
30000000
03000000
00100000
00010000
00004400
00000100
00000001
00000010

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

C24.378C23 in GAP, Magma, Sage, TeX 

C_2^4._{378}C_2^3
 
% in TeX
 
G:=Group("C2^4.378C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1395);
 
// by ID
 
G=gap.SmallGroup(128,1395);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,185,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=d*b=b*d,g^2=c,e*a*e^-1=a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,a*g=g*a,b*c=c*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=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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