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G = C24.230C23  order 128 = 27

70th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.230C23, C23.265C24, C22.702- 1+4, C22.962+ 1+4, C42⋊33(C2×C4), C42⋊2C2⋊3C4, C42⋊5C4⋊9C2, C42⋊9C4⋊14C2, C23.31(C22×C4), (C23×C4).64C22, (C22×C4).493C23, (C2×C42).453C22, C22.156(C23×C4), C23.8Q8.10C2, C23.63C23⋊27C2, C23.65C23⋊32C2, C2.5(C22.54C24), C2.C42.73C22, C24.C22.10C2, C2.6(C22.57C24), C2.43(C23.33C23), C4⋊C4⋊21(C2×C4), C22⋊C4.14(C2×C4), (C2×C4).61(C22×C4), (C2×C4⋊C4).200C22, (C2×C42⋊2C2).4C2, (C2×C22⋊C4).45C22, SmallGroup(128,1115)

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

Derived series
C1 — C22 — C24.230C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C42⋊2C2 — C24.230C23
Lower central
C1 — C22 — C24.230C23
Upper central
C1 — C23 — C24.230C23
Jennings
C1 — C23 — C24.230C23

Generators and relations for C24.230C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=c, f2=a, g2=b, ab=ba, ac=ca, ede=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, gdg-1=abd, fef-1=abe, fg=gf >

Subgroups: 396 in 228 conjugacy classes, 132 normal (13 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊2C2, C23×C4, C42⋊5C4, C42⋊9C4, C23.8Q8, C23.63C23, C24.C22, C23.65C23, C2×C42⋊2C2, C24.230C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2+ 1+4, 2- 1+4, C23.33C23, C22.54C24, C22.57C24, C24.230C23

Smallest permutation representation of C24.230C23
►On 64 points
Generators in S64
(1 11)(2 12)(3 9)(4 10)(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 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 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)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4AB
order12···2224···4
size11···1444···4

38 irreducible representations

dim11111111144
type+++++++++-
imageC1C2C2C2C2C2C2C2C42+ 1+42- 1+4
kernelC24.230C23C42⋊5C4C42⋊9C4C23.8Q8C23.63C23C24.C22C23.65C23C2×C42⋊2C2C42⋊2C2C22C22
# reps111333311633

Matrix representation of C24.230C23 ►in GL9(𝔽5)

100000000
040000000
004000000
000400000
000040000
000001000
000000100
000000010
000000001
,
100000000
010000000
001000000
000100000
000010000
000004000
000000400
000000040
000000004
,
400000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
000000004
,
200000000
012000000
044000000
004040000
011100000
000002030
000000203
000000030
000000003
,
400000000
010000000
044000000
040400000
000010000
000001000
000000400
000000010
000000004
,
100000000
040300000
000110000
010100000
044400000
000000100
000001000
000000204
000002040
,
100000000
043000000
001000000
001010000
004100000
000003000
000000300
000001020
000000102

G:=sub<GL(9,GF(5))| [1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,0,1,4,0,1,0,0,0,0,0,2,4,4,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,3,0,3],[4,0,0,0,0,0,0,0,0,0,1,4,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,4,0,1,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,1,1,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0],[1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,1,1,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2] >;
 

C24.230C23 in GAP, Magma, Sage, TeX 

C_2^4._{230}C_2^3
 
% in TeX
 
G:=Group("C2^4.230C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1115);
 
// by ID
 
G=gap.SmallGroup(128,1115);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,555,100,1571,346,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=c,f^2=a,g^2=b,a*b=b*a,a*c=c*a,e*d*e=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^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g^-1=a*b*d,f*e*f^-1=a*b*e,f*g=g*f>;
 
// generators/relations
 

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