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G = C24.554C23  order 128 = 27

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

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

Aliases: C24.554C23, C23.227C24, 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), (C22×C4).756C23, (C2×C42).428C22, (C23×C4).301C22, C22.118(C23×C4), C23.217(C22×C4), C23.7Q8.30C2, C23.65C23⋊19C2, C23.63C23⋊14C2, C2.25(C22.11C24), C2.C42.473C22, C2.1(C22.47C24), (C4×C4⋊C4)⋊33C2, (C2×C4⋊C4)⋊37C4, 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, (C2×C4).232(C22×C4), (C22×C4).310(C2×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 — C24.554C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C4⋊C4 — C24.554C23
Lower central
C1 — C22 — C24.554C23
Upper central
C1 — C23 — C24.554C23
Jennings
C1 — C23 — C24.554C23

Subgroups: 444 in 274 conjugacy classes, 156 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×20], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×20], C2×C4 [×48], C23, C23 [×6], C23 [×4], C42 [×8], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×16], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×24], C22×C4 [×8], C24, C2.C42 [×12], C2×C42 [×6], C2×C22⋊C4 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×16], C23×C4, C23×C4 [×2], C4×C22⋊C4, C4×C22⋊C4 [×2], C4×C4⋊C4 [×2], C23.7Q8, C23.7Q8 [×2], C23.63C23 [×4], C23.65C23 [×2], C22×C4⋊C4, C24.554C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×6], C24, C42⋊C2 [×4], C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4 [×3], 2+ (1+4) [×2], C2×C42⋊C2, C2×C4×Q8, C22.11C24, C22.47C24 [×2], D4⋊3Q8 [×2], C24.554C23

Generators and relations
 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 >

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

(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)

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

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

Matrix representation ►G ⊆ 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] >;
 

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)
kernelC24.554C23C4×C22⋊C4C4×C4⋊C4C23.7Q8C23.63C23C23.65C23C22×C4⋊C4C2×C4⋊C4C22⋊C4C2×C4C23C22
# reps1323421164842

In GAP, Magma, Sage, TeX 

C_2^4._{554}C_2^3
 
% in TeX
 
G:=Group("C2^4.554C2^3");
 
// 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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