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G = C23.548C24  order 128 = 27

265th central stem extension by C23 of C24

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

Aliases: C24.46C23, C23.548C24, C22.3232+ (1+4), C23.70(C4○D4), C23⋊2D4.19C2, (C2×C42).80C22, C23.Q8⋊43C2, C23.10D4⋊65C2, C23.11D4⋊67C2, C23.23D4⋊74C2, (C23×C4).143C22, (C22×C4).158C23, C24.3C22⋊68C2, (C22×D4).202C22, C24.C22⋊107C2, C23.83C23⋊66C2, C2.46(C22.32C24), C23.63C23⋊115C2, C2.C42.557C22, C2.32(C22.34C24), C2.100(C23.36C23), (C4×C22⋊C4)⋊95C2, (C2×C4).173(C4○D4), (C2×C4⋊C4).374C22, C22.420(C2×C4○D4), (C2×C22⋊C4).471C22, SmallGroup(128,1380)

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

Derived series
C1 — C23 — C23.548C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C23.23D4 — C23.548C24
Lower central
C1 — C23 — C23.548C24
Upper central
C1 — C23 — C23.548C24
Jennings
C1 — C23 — C23.548C24

Subgroups: 548 in 242 conjugacy classes, 88 normal (82 characteristic)
C1, C2 [×7], C2 [×4], C4 [×14], C22 [×7], C22 [×24], C2×C4 [×4], C2×C4 [×38], D4 [×12], C23, C23 [×2], C23 [×20], C42 [×2], C22⋊C4 [×18], C4⋊C4 [×6], C22×C4 [×12], C22×C4 [×4], C2×D4 [×12], C24 [×3], C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×14], C2×C4⋊C4 [×5], C23×C4, C22×D4 [×3], C4×C22⋊C4, C23.23D4 [×2], C23.63C23, C24.C22 [×2], C24.3C22, C23⋊2D4, C23.10D4 [×3], C23.Q8, C23.11D4 [×2], C23.83C23, C23.548C24

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ (1+4) [×4], C23.36C23, C22.32C24 [×4], C22.34C24 [×2], C23.548C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=c, g2=b, ab=ba, ac=ca, ede=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
02000000
20000000
00300000
00030000
00001000
00000100
00004040
00004004
,
01000000
10000000
00130000
00040000
00004030
00000041
00000010
00000110
,
40000000
01000000
00100000
00140000
00001000
00001400
00000010
00004004
,
30000000
03000000
00100000
00010000
00001300
00000400
00000101
00000110

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4N4O···4T
order12···2222244444···44···4
size11···1448822224···48···8

32 irreducible representations

dim11111111111224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C4○D4C4○D42+ (1+4)
kernelC23.548C24C4×C22⋊C4C23.23D4C23.63C23C24.C22C24.3C22C23⋊2D4C23.10D4C23.Q8C23.11D4C23.83C23C2×C4C23C22
# reps11212113121844

In GAP, Magma, Sage, TeX 

C_2^3._{548}C_2^4
 
% in TeX
 
G:=Group("C2^3.548C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1380);
 
// by ID
 
G=gap.SmallGroup(128,1380);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,185,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=c,g^2=b,a*b=b*a,a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,g*f*g^-1=a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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