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G = C24.377C23  order 128 = 27

217th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.377C23, C23.561C24, C22.3352+ (1+4), (C22×C4)⋊39D4, C23⋊2D4⋊30C2, C23.200(C2×D4), C23.10D4⋊66C2, C23.11D4⋊71C2, C2.33(C23⋊3D4), (C22×C4).166C23, (C23×C4).437C22, (C2×C42).625C22, C22.373(C22×D4), C24.3C22⋊69C2, (C22×D4).209C22, C24.C22⋊111C2, C23.81C23⋊72C2, C2.49(C22.29C24), C2.52(C22.32C24), C2.C42.275C22, C2.50(C22.26C24), C2.36(C22.34C24), (C2×C4⋊D4)⋊26C2, (C4×C22⋊C4)⋊98C2, (C2×C4).683(C2×D4), (C2×C4).181(C4○D4), (C2×C4⋊C4).384C22, C22.428(C2×C4○D4), (C2×C22.D4)⋊27C2, (C2×C22⋊C4).520C22, SmallGroup(128,1393)

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

Derived series
C1 — C23 — C24.377C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C24.3C22 — C24.377C23
Lower central
C1 — C23 — C24.377C23
Upper central
C1 — C23 — C24.377C23
Jennings
C1 — C23 — C24.377C23

Subgroups: 692 in 299 conjugacy classes, 96 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×5], C4 [×15], C22 [×3], C22 [×4], C22 [×31], C2×C4 [×8], C2×C4 [×33], D4 [×24], C23, C23 [×2], C23 [×27], C42 [×2], C22⋊C4 [×24], C4⋊C4 [×8], C22×C4 [×5], C22×C4 [×10], C22×C4 [×2], C2×D4 [×24], C24 [×2], C24 [×2], C2.C42 [×2], C2.C42 [×4], C2×C42 [×2], C2×C22⋊C4 [×3], C2×C22⋊C4 [×12], C2×C4⋊C4 [×3], C2×C4⋊C4 [×2], C4⋊D4 [×4], C22.D4 [×4], C23×C4, C22×D4 [×2], C22×D4 [×4], C4×C22⋊C4, C24.C22 [×2], C24.3C22 [×2], C23⋊2D4, C23⋊2D4 [×2], C23.10D4, C23.10D4 [×2], C23.11D4, C23.81C23, C2×C4⋊D4, C2×C22.D4, C24.377C23

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) [×4], C22.26C24, C23⋊3D4, C22.29C24, C22.32C24 [×2], C22.34C24 [×2], C24.377C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=d, g2=c, eae-1=ab=ba, faf=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=de=ed, df=fd, dg=gd, eg=ge >

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
01000000
00230000
00430000
00000010
00000001
00001000
00000100
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
12000000
44000000
00100000
00010000
00004000
00000400
00000010
00000001
,
43000000
01000000
00100000
00240000
00004000
00002100
00000040
00000021
,
10000000
01000000
00200000
00020000
00002200
00001300
00000022
00000013

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A4B4C4D4E···4N4O···4S
order12···22222244444···44···4
size11···14488822224···48···8

32 irreducible representations

dim1111111111224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C4○D42+ (1+4)
kernelC24.377C23C4×C22⋊C4C24.C22C24.3C22C23⋊2D4C23.10D4C23.11D4C23.81C23C2×C4⋊D4C2×C22.D4C22×C4C2×C4C22
# reps1122331111484

In GAP, Magma, Sage, TeX 

C_2^4._{377}C_2^3
 
% in TeX
 
G:=Group("C2^4.377C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1393);
 
// by ID
 
G=gap.SmallGroup(128,1393);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,723,185,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=d,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*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=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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