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G = C24.378C23order 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⋊Q836C2, (C22×C4).409D4, C23.201(C2×D4), C23.4Q837C2, C23.11D472C2, C2.34(C233D4), (C2×C42).627C22, (C22×C4).168C23, (C23×C4).438C22, C22.375(C22×D4), C23.10D4.37C2, (C22×D4).211C22, (C22×Q8).169C22, C23.78C2334C2, C23.83C2371C2, C24.C22113C2, C23.65C23110C2, 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

C1C23 — C24.378C23
C1C2C22C23C22×C4C2×C22⋊C4C24.C22 — C24.378C23
C1C23 — C24.378C23
C1C23 — C24.378C23
C1C23 — 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 [×3], C2 [×4], C2 [×3], C4 [×17], C22 [×3], C22 [×4], C22 [×17], C2×C4 [×8], C2×C4 [×39], D4 [×4], Q8 [×4], C23, C23 [×2], C23 [×13], C42 [×2], C22⋊C4 [×20], C4⋊C4 [×16], C22×C4 [×5], C22×C4 [×12], C22×C4 [×2], C2×D4 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×3], C2×C22⋊C4 [×8], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C22⋊Q8 [×4], C22.D4 [×4], C23×C4, C22×D4, C22×Q8, C4×C22⋊C4, C24.C22 [×2], C23.65C23 [×2], C23⋊Q8, C23.10D4 [×2], C23.78C23, C23.11D4 [×2], C23.4Q8, C23.83C23, C2×C22⋊Q8, C2×C22.D4, C24.378C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C22.26C24, C233D4, C23.38C23, C22.33C24 [×2], C22.36C24 [×2], 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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