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G = C23.227C24  order 128 = 27

80th central extension by C23 of C24

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

Aliases: C23.227C24, C24.554C23, C22.632+ 1+4, C22⋊C4.11Q8, C22.16(C4×Q8), C2.1(D4⋊3Q8), C23.112(C2×Q8), C4.27(C42⋊C2), C23.322(C4○D4), C22.37(C22×Q8), (C2×C42).428C22, (C23×C4).301C22, C22.118(C23×C4), C23.217(C22×C4), (C22×C4).756C23, C23.7Q8.30C2, C23.65C23⋊19C2, C23.63C23⋊14C2, C2.25(C22.11C24), C2.C42.473C22, C2.1(C22.47C24), (C2×C4⋊C4)⋊37C4, (C4×C4⋊C4)⋊33C2, C2.15(C2×C4×Q8), C22⋊C4○4(C4⋊C4), C4⋊C4.207(C2×C4), (C2×C4).251(C2×Q8), (C22×C4⋊C4).28C2, (C4×C22⋊C4).26C2, (C2×C4).790(C4○D4), (C2×C4⋊C4).185C22, (C22×C4).310(C2×C4), (C2×C4).232(C22×C4), C2.28(C2×C42⋊C2), C22.112(C2×C4○D4), (C2×C22⋊C4).437C22, SmallGroup(128,1077)

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

Derived series
C1 — C22 — C23.227C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C4⋊C4 — C23.227C24
Lower central
C1 — C22 — C23.227C24
Upper central
C1 — C23 — C23.227C24
Jennings
C1 — C23 — C23.227C24

Generators and relations for C23.227C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=cb=bc, g2=b, ab=ba, eae-1=ac=ca, ad=da, af=fa, ag=ga, bd=db, fef-1=be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 444 in 274 conjugacy classes, 156 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C23×C4, C4×C22⋊C4, C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C23.7Q8, C23.63C23, C23.65C23, C22×C4⋊C4, C23.227C24
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, C24, C42⋊C2, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, 2+ 1+4, C2×C42⋊C2, C2×C4×Q8, C22.11C24, C22.47C24, D4⋊3Q8, C23.227C24

Smallest permutation representation of C23.227C24
►On 64 points
Generators in S64
(1 3)(2 50)(4 52)(5 64)(6 8)(7 62)(9 11)(10 22)(12 24)(13 15)(14 26)(16 28)(17 19)(18 30)(20 32)(21 23)(25 27)(29 31)(33 35)(34 38)(36 40)(37 39)(41 43)(42 54)(44 56)(45 47)(46 58)(48 60)(49 51)(53 55)(57 59)(61 63)

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4T4U···4AL
order12···222224···44···4
size11···122222···24···4

50 irreducible representations

dim111111112224
type+++++++-+
imageC1C2C2C2C2C2C2C4Q8C4○D4C4○D42+ 1+4
kernelC23.227C24C4×C22⋊C4C4×C4⋊C4C23.7Q8C23.63C23C23.65C23C22×C4⋊C4C2×C4⋊C4C22⋊C4C2×C4C23C22
# reps1323421164842

Matrix representation of C23.227C24 ►in GL5(𝔽5)

10000
01400
00400
00010
00001
,
10000
01000
00100
00040
00004
,
10000
04000
00400
00010
00001
,
40000
01000
00100
00010
00001
,
30000
01000
02400
00021
00023
,
40000
03000
00300
00030
00032
,
10000
04000
00400
00042
00041

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

C23.227C24 in GAP, Magma, Sage, TeX 

C_2^3._{227}C_2^4
 
% in TeX
 
G:=Group("C2^3.227C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1077);
 
// by ID
 
G=gap.SmallGroup(128,1077);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=c*b=b*c,g^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,a*f=f*a,a*g=g*a,b*d=d*b,f*e*f^-1=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,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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