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G = C23.603C24  order 128 = 27

320th central stem extension by C23 of C24

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

Aliases: C24.64C23, C23.603C24, C22.3772+ 1+4, C22⋊C4⋊16D4, C23⋊2D4⋊42C2, C23.217(C2×D4), C23.80(C4○D4), C2.108(D4⋊5D4), C23.7Q8⋊92C2, C23.23D4⋊92C2, C23.11D4⋊89C2, C23.10D4⋊88C2, (C2×C42).655C22, (C23×C4).151C22, (C22×C4).185C23, C22.412(C22×D4), C24.3C22⋊83C2, (C22×D4).239C22, C24.C22⋊134C2, C23.83C23⋊83C2, C2.59(C22.29C24), C2.69(C22.32C24), C2.17(C22.54C24), C2.C42.309C22, C2.43(C22.34C24), C2.85(C22.47C24), (C2×C4⋊D4)⋊37C2, (C2×C4).105(C2×D4), (C2×C4).429(C4○D4), (C2×C4⋊C4).416C22, C22.465(C2×C4○D4), (C2×C22.D4)⋊38C2, (C2×C22⋊C4).269C22, SmallGroup(128,1435)

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

Derived series
C1 — C23 — C23.603C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23.7Q8 — C23.603C24
Lower central
C1 — C23 — C23.603C24
Upper central
C1 — C23 — C23.603C24
Jennings
C1 — C23 — C23.603C24

Generators and relations for C23.603C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=b, e2=ba=ab, g2=a, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

Subgroups: 708 in 306 conjugacy classes, 96 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C22.D4, C23×C4, C22×D4, C23.7Q8, C23.23D4, C24.C22, C24.3C22, C23⋊2D4, C23.10D4, C23.11D4, C23.83C23, C2×C4⋊D4, C2×C22.D4, C23.603C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C22.29C24, C22.32C24, C22.34C24, D4⋊5D4, C22.47C24, C22.54C24, C23.603C24

Smallest permutation representation of C23.603C24
►On 64 points
Generators in S64
(1 37)(2 38)(3 39)(4 40)(5 11)(6 12)(7 9)(8 10)(13 41)(14 42)(15 43)(16 44)(17 46)(18 47)(19 48)(20 45)(21 50)(22 51)(23 52)(24 49)(25 56)(26 53)(27 54)(28 55)(29 60)(30 57)(31 58)(32 59)(33 64)(34 61)(35 62)(36 63)

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4N4O4P4Q4R
order12···22222224···44444
size11···14444884···48888

32 irreducible representations

dim111111111112224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC23.603C24C23.7Q8C23.23D4C24.C22C24.3C22C23⋊2D4C23.10D4C23.11D4C23.83C23C2×C4⋊D4C2×C22.D4C22⋊C4C2×C4C23C22
# reps113213111114444

Matrix representation of C23.603C24 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
220000
130000
002000
000300
000010
000001
,
220000
030000
003000
000300
000010
000004
,
100000
010000
000100
001000
000001
000010
,
300000
420000
000100
001000
000010
000001

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,1,0,0,0,0,2,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,2,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,4,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C23.603C24 in GAP, Magma, Sage, TeX 

C_2^3._{603}C_2^4
 
% in TeX
 
G:=Group("C2^3.603C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1435);
 
// by ID
 
G=gap.SmallGroup(128,1435);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,344,758,723,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=b,e^2=b*a=a*b,g^2=a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-1=a*e=e*a,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,g*d*g^-1=a*b*d,f*g=g*f>;
 
// generators/relations
 

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