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G = C24.195C23order 128 = 27

35th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.195C23, C23.204C24, C22.432+ 1+4, C22.272- 1+4, C22.43(C4×D4), C23.607(C2×D4), (C22×C4).167D4, C22.D47C4, C23.8Q86C2, C2.4(C233D4), (C23×C4).45C22, C22.95(C23×C4), C22.92(C22×D4), C23.224(C4○D4), C23.34D412C2, C23.125(C22×C4), (C22×C4).469C23, C24.C225C2, (C2×C42).411C22, C23.23D4.4C2, C23.63C234C2, (C22×D4).477C22, C23.65C2312C2, C2.C42.40C22, C2.3(C22.33C24), C2.4(C23.38C23), C2.11(C23.33C23), C4⋊C49(C2×C4), C2.21(C2×C4×D4), (C2×C4×D4).31C2, C22⋊C49(C2×C4), (C22×C4⋊C4)⋊8C2, (C4×C22⋊C4)⋊32C2, (C22×C4)⋊23(C2×C4), (C2×C4).676(C2×D4), (C2×D4).167(C2×C4), (C2×C4).25(C22×C4), C22.89(C2×C4○D4), (C2×C4⋊C4).177C22, (C2×C22.D4).5C2, (C2×C22⋊C4).426C22, SmallGroup(128,1054)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.195C23
C1C2C22C23C24C23×C4C22×C4⋊C4 — C24.195C23
C1C22 — C24.195C23
C1C23 — C24.195C23
C1C23 — C24.195C23

Generators and relations for C24.195C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=g2=b, eae-1=gag-1=ab=ba, ac=ca, ad=da, faf-1=abc, bc=cb, bd=db, fef-1=geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, gfg-1=bcf >

Subgroups: 556 in 316 conjugacy classes, 148 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×20], C22 [×3], C22 [×8], C22 [×22], C2×C4 [×14], C2×C4 [×52], D4 [×8], C23, C23 [×8], C23 [×10], C42 [×4], C22⋊C4 [×12], C22⋊C4 [×8], C4⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×5], C22×C4 [×16], C22×C4 [×14], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×10], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×4], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×8], C2×C4⋊C4 [×4], C4×D4 [×4], C22.D4 [×8], C23×C4 [×4], C22×D4, C4×C22⋊C4, C23.34D4, C23.8Q8, C23.8Q8 [×2], C23.23D4, C23.63C23 [×2], C24.C22 [×2], C23.65C23 [×2], C22×C4⋊C4, C2×C4×D4, C2×C22.D4, C24.195C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, 2+ 1+4 [×2], 2- 1+4 [×2], C2×C4×D4, C23.33C23 [×2], C233D4, C23.38C23, C22.33C24 [×2], C24.195C23

Smallest permutation representation of C24.195C23
On 64 points
Generators in S64
(2 13)(4 15)(5 64)(6 38)(7 62)(8 40)(10 35)(12 33)(17 50)(19 52)(21 27)(22 57)(23 25)(24 59)(26 41)(28 43)(30 55)(32 53)(37 46)(39 48)(42 60)(44 58)(45 63)(47 61)
(1 16)(2 13)(3 14)(4 15)(5 46)(6 47)(7 48)(8 45)(9 34)(10 35)(11 36)(12 33)(17 50)(18 51)(19 52)(20 49)(21 42)(22 43)(23 44)(24 41)(25 58)(26 59)(27 60)(28 57)(29 54)(30 55)(31 56)(32 53)(37 64)(38 61)(39 62)(40 63)
(1 20)(2 17)(3 18)(4 19)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 29)(12 30)(13 50)(14 51)(15 52)(16 49)(21 27)(22 28)(23 25)(24 26)(33 55)(34 56)(35 53)(36 54)(41 59)(42 60)(43 57)(44 58)(45 63)(46 64)(47 61)(48 62)
(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 43 16 22)(2 23 13 44)(3 41 14 24)(4 21 15 42)(5 34 46 9)(6 10 47 35)(7 36 48 11)(8 12 45 33)(17 25 50 58)(18 59 51 26)(19 27 52 60)(20 57 49 28)(29 39 54 62)(30 63 55 40)(31 37 56 64)(32 61 53 38)
(1 35 16 10)(2 11 13 36)(3 33 14 12)(4 9 15 34)(5 27 46 60)(6 57 47 28)(7 25 48 58)(8 59 45 26)(17 29 50 54)(18 55 51 30)(19 31 52 56)(20 53 49 32)(21 64 42 37)(22 38 43 61)(23 62 44 39)(24 40 41 63)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4AD
order12···22222224···44···4
size11···12222442···24···4

44 irreducible representations

dim1111111111112244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4D4C4○D42+ 1+42- 1+4
kernelC24.195C23C4×C22⋊C4C23.34D4C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C22×C4⋊C4C2×C4×D4C2×C22.D4C22.D4C22×C4C23C22C22
# reps11131222111164422

Matrix representation of C24.195C23 in GL8(𝔽5)

10000000
04000000
00100000
00040000
00001000
00000400
00000040
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
20000000
02000000
00100000
00010000
00000040
00000004
00004000
00000400
,
01000000
10000000
00010000
00100000
00000200
00002000
00000003
00000030
,
40000000
01000000
00400000
00010000
00000100
00004000
00000004
00000010

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

C24.195C23 in GAP, Magma, Sage, TeX

C_2^4._{195}C_2^3
% in TeX

G:=Group("C2^4.195C2^3");
// GroupNames label

G:=SmallGroup(128,1054);
// by ID

G=gap.SmallGroup(128,1054);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,100,675]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=g^2=b,e*a*e^-1=g*a*g^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c,b*c=c*b,b*d=d*b,f*e*f^-1=g*e*g^-1=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,d*e=e*d,d*f=f*d,d*g=g*d,g*f*g^-1=b*c*f>;
// generators/relations

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