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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, C23.201(C2×D4), (C22×C4).409D4, C23.4Q8⋊37C2, C23.11D4⋊72C2, C2.34(C23⋊3D4), (C23×C4).438C22, (C2×C42).627C22, (C22×C4).168C23, C22.375(C22×D4), C23.10D4.37C2, (C22×D4).211C22, (C22×Q8).169C22, C24.C22⋊113C2, C23.78C23⋊34C2, C23.83C23⋊71C2, C23.65C23⋊110C2, C2.C42.277C22, C2.52(C22.26C24), C2.53(C22.33C24), C2.49(C23.38C23), C2.64(C22.36C24), (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

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, C23⋊3D4, C23.38C23, C22.33C24 [×2], C22.36C24 [×2], C24.378C23

Generators and relations
 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 >

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

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

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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+4)2- (1+4)
kernelC24.378C23C4×C22⋊C4C24.C22C23.65C23C23⋊Q8C23.10D4C23.78C23C23.11D4C23.4Q8C23.83C23C2×C22⋊Q8C2×C22.D4C22×C4C2×C4C22C22
# reps1122121211114822

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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