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G = C23.334C24  order 128 = 27

51st central stem extension by C23 of C24

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

Aliases: C23.334C24, C24.564C23, C22.1052- 1+4, C2.10(D4×Q8), C22⋊C4⋊12Q8, C22⋊3(C4⋊Q8), C4.30C22≀C2, (C2×Q8).222D4, (Q8×C23).8C2, C23.609(C2×D4), (C22×C4).377D4, C23.117(C2×Q8), C22.67(C22×Q8), (C23×C4).347C22, (C22×C4).801C23, (C2×C42).479C22, C22.214(C22×D4), C23.8Q8.11C2, C23.78C23⋊6C2, (C22×Q8).423C22, C23.67C23⋊37C2, C2.C42.94C22, C2.13(C23.38C23), (C2×C4⋊Q8)⋊6C2, (C2×C4)⋊2(C2×Q8), C2.8(C2×C4⋊Q8), (C2×C4).318(C2×D4), C2.22(C2×C22≀C2), (C4×C22⋊C4).38C2, (C2×C22⋊Q8).25C2, (C2×C4⋊C4).219C22, (C2×C22⋊C4).492C22, SmallGroup(128,1166)

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

Derived series
C1 — C23 — C23.334C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C23.334C24
Lower central
C1 — C23 — C23.334C24
Upper central
C1 — C23 — C23.334C24
Jennings
C1 — C23 — C23.334C24

Generators and relations for C23.334C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=b, f2=c, ab=ba, ac=ca, faf-1=ad=da, ae=ea, ag=ga, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd >

Subgroups: 676 in 400 conjugacy classes, 140 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C4⋊Q8, C23×C4, C23×C4, C22×Q8, C22×Q8, C22×Q8, C4×C22⋊C4, C23.8Q8, C23.67C23, C23.78C23, C2×C22⋊Q8, C2×C4⋊Q8, Q8×C23, C23.334C24
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C22≀C2, C4⋊Q8, C22×D4, C22×Q8, 2- 1+4, C2×C22≀C2, C2×C4⋊Q8, C23.38C23, D4×Q8, C23.334C24

Smallest permutation representation of C23.334C24
►On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 24)(6 21)(7 22)(8 23)(13 41)(14 42)(15 43)(16 44)(17 61)(18 62)(19 63)(20 64)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 46)(34 47)(35 48)(36 45)(37 50)(38 51)(39 52)(40 49)

(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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(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 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4V4W4X4Y4Z
order12···2222244444···44444
size11···1222222224···48888

38 irreducible representations

dim111111112224
type++++++++-++-
imageC1C2C2C2C2C2C2C2Q8D4D42- 1+4
kernelC23.334C24C4×C22⋊C4C23.8Q8C23.67C23C23.78C23C2×C22⋊Q8C2×C4⋊Q8Q8×C23C22⋊C4C22×C4C2×Q8C22
# reps114241218482

Matrix representation of C23.334C24 ►in GL6(𝔽5)

400000
040000
001000
000100
000040
000041
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
300000
020000
002100
000300
000010
000014
,
010000
400000
001000
000100
000013
000014
,
010000
400000
001300
001400
000010
000014

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

C23.334C24 in GAP, Magma, Sage, TeX 

C_2^3._{334}C_2^4
 
% in TeX
 
G:=Group("C2^3.334C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1166);
 
// by ID
 
G=gap.SmallGroup(128,1166);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=b,f^2=c,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*a,a*g=g*a,b*c=c*b,b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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